X-ray study of the roughening transition of the Ni(113) surface

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X-ray study of the roughening transition of the Ni(113)

surface

I.K. Robinson, E.H. Conrad, D.S. Reed

To cite this version:

X-ray

study

of the

_roughening

transition of the

_Ni(113)

surface

I. K. Robinson

_(1),

E. H. Conrad

₍₂₎

and D. S. Reed

₍₂₎

(1)

AT & T Bell

_{Laboratories, Murray}

Hill, New

_{Jersey 07974,}

U.S.A.

(2)

Dept.

of

_Physics

&

_Astronomy,

Univ. of

Missouri,

Columbia MO _65211, U.S.A.

(Reçu

le 9 mai 1989, révisé le 28 août 1989,

accepté

le 19

septembre

1989)

Résumé. 2014 Nous

présentons

des résultats de diffraction X en fonction de la

_température

pour

Ni(113)

qui

montrent des

_{changements spectaculaires}

des « bâtonnets de terminaison ». La variation de l’intensité est en bon accord avec les travaux antérieurs sur

Cu(113)

mais nous

pouvons maintenant résoudre aussi des

changements

dans la forme des

pics.

Nous

développons

une théorie de la diffraction des surfaces rugueuses et nous _{l’utilisons pour} montrer _{que la surface}

a des fluctuations finies

d’amplitudes

au-dessous de la transition

qui

deviennent

_divergentes

au-dessus. La forme fonctionnelle de la

_divergence

ne _peut

_cependant

_{pas être évaluée de}

_façon

unique.

Abstract. 2014 We

present

X-ray

diffraction data as a function of temperature for

_Ni(113)

that show dramatic

_changes

in one of the

_crystal

truncation rods. The variation of the observed

intensity

is in

good

agreement with

_previous

work on

Cu(113),

but we can now resolve

_changes

of

_lineshape

as

well. We

_develop

a

_theory

of diffraction of

rough

surfaces and use it to show

conclusively

that the surface has finite

_height

fluctuations below the transition that become

_divergent

above. The exact functional form of the

divergence

cannot be evaluated

_uniquely,

however.

Classification

Physics

Abstracts 61.50J - 61.10 - 68.20

The

_stability

of a surface can be

_predicted

from its

_microscopic

interactions

_by

the well known

Wulff construction

_[1].

Low-index surfaces are often stable at all

_temperatures

at which the solid

exists,

but

_high-index

surfaces

_normally

_{have lower free energy in} a

_rough

state

_whereby

the

_{configurational}

entropy

term exceeds the

_potential

_{energy. An} intermediate case however

can be smooth at low

_temperature

and

_rough

at

_high

_temperature,

_{thereby having}

a

_phase

transition at the

_roughening

_temperature,

_TR,

first

_{suggested by}

Burton and Cabrera

_[2].

This is known to occur in electron

_microscope

studies of the

_{equilibrium crystal shape}

of small Pb

and Cu

_crystals

_[3],

which

_present

different facets at different

_{temperatures.}

_Recently

there have been studies of the

_stability

of extended

_crystal

faces

_by

He and

_X-ray

diffraction that also show a well-defined transition.

_Agreement

with the statistical mechanical

_theory

for T «

_TR

due to Villain et al.

_[4],

has been

_reported

in

_X-ray

studies of

_{Cu(110) [5]}

and

_Cu(113)

[6],

while He diffraction work on

_{Ni(115) [7], Ni(113) [8], Cu(115) [9]}

and

X-ray

work on

Ag(110) [10]

have been

_interpreted

in terms of a

_{logarithmic divergence}

of the

_height

fluctuations valid for T>

_TR

as

_{originally suggested by}

Chui and Weeks

_[11].

We will discuss both these theories in _{this paper.}

Our

_experiments

use

_X-ray

diffraction because the Born

_{approximation}

ensures that it is

accurately

kinematical. The diffracted

_amplitude

is

_just

a

_{superposition}

of scattered

contributions from all

_parts

of the

_sample.

Here lies the main

_advantage

over He and electron

diffraction which are

_inherently

much more surface sensitive. In this work

_grazing

incident/exit

_angle

conditions were used to reduce

_{background ;}

a kinematic

_description

of the

diffraction is still

_appropriate

however

_[12].

Surface

_sensitivity

in

_X-ray

diffraction is obtained

by using

an intense source

(synchrotron

radiation from an electron

_storage

ring)

and

_selecting,

by

symmetry

only,

the

_parts

of the diffraction

_pattern

due to the surface. Because a

_relatively

small total momentum transfer is

used,

phonon

effects are

_kept

to a minimum. The

_X-ray

surface

_technique

has been

_widely

used over the last few years for surface structure

determination

_[13]

and

_recently

for

_{studying phase}

transitions

_[5,

6,

10,

14].

Before

_presenting

the data for

_Ni(113)

we will write down the

theory

of

_X-ray

diffraction

from

_rough

surfaces and show how it

_predicts

the

_lineshapes

for the two

_roughening

theories mentioned above. This section is based on the

_theory

of

crystal

truncation rods

[15]

and is

closely analogous

to the calculation of He diffraction from

_rough

surfaces

_[4]

with some

important

differences. It also has some

_similarity

to the calculation of Mochrie

_[5].

Diffraction from

rough

surfaces.

We assume a Solid on Solid

_(SOS)

model for the

_crystal

with a

_rough

surface

[16]

which has all

atoms on an ideal cubic lattice truncated

_by

a

_height

function

_{h (x, y )}

as shown in

_figure

la. We use dimensionless

_{crystallographic}

coordinates such that x and y

_represent

the number of

unit cells

_along

the

_in-plane

directions and z is normal to the surface. The

_components

of

momentum transfer are

_similarly

dimensionless,

_(qx,

_qy,

_qz).

The

_{kinematically}

scattered

amplitude

from this

_{arrangement is,}

The summation of x

_implicitly

assumes a sum in y as well. The second sum in

equation

(2)

gives

rise to the

_crystal

truncation rod in

_{the qz}

direction

_{perpendicular}

to the surface. For mathematical

_completeness

_{it is necessary} to add a small

_imaginary

_part

to _{qz, which} attenuates the contributions from

_{deeper layers.}

This term is _{then let go} to zero

_[15].

This

Fig.

1. -

(a)

The solid on solid

_(SOS)

model of a

_rough

surface of a cubic

_crystal.

The

_boundary

is

represented by

a

_single

valued function

h (x )

which is discrete. No

overhangs

are allowed therefore.

_(b)

Sectional view of a face-centered cubic

(f.c.c.)

113 surface. Axes for x and z,

respectively

parallel

and

corresponds

to the

_physical

situation of

_absorption

of the incident

_X-ray

beam. The result for all

_points

on the

_rod,

_except

the

_{Bragg points qz}

= 2 _7Tn, is

[15]

The diffracted

_intensity

7 is

_proportional

to

_{(AA *)}

_{ensemble-averaged}

over all

_height

configurations

The surface is assumed to be the same

_everywhere

so we can use the

_stationary

_property

The ensemble average in

equation (5)

is evaluated in reference

_[4]

in terms of the

height-height

correlation

function,

which is the

_quantity

_{predicted by}

the statistical mechanical models we use. The

_{approximations}

used

_[4]

are _{poor for the} case

_{of qz =}

_{(2 n}

+

₁₎

7T which is

the case most relevant to _{this paper,} as well as most other

_experimental

work,

so we will

derive a new

_expression

for the ensemble average in

_equation

_(5).

We assume that the

_height

of the surface is a discrete Gaussian random variable with a distribution

where a

depends

on

_e.

_{The ensemble average} we

_require

is then

We will assume that the

_height

distribution is broad so that a « 1 and the summation

_{in p}

is

dominated

_by

its central term. We can also substitute

where

_[qz]

means qz modulo 2 7T such that - 7T

[qz]

7T. The case of interest

[qz]

= 7T will

_require

an extra factor of 2 in the

_intensity

because two terms are

_equally

important

in the sum over _{p, but} we will

_ignore

this. The

_resulting

formula for the

_intensity

is :

This is the fundamental

_equation

for

_describing

the diffraction of

_rough

surfaces in the SOS model. It contains two

_factors,

an ideal

_1/sin’

_(qz/2)

_crystal

truncation rod

_[15]

_(CTR)

intensity

dependence

on the

_{perpendicular}

momentum _{transfer qz,}

_{multiplied by}

the Fourier transform of the

_pair

correlation function

_C(e)

_parallel

to the surface :

Figure

2 shows some

_{graphic examples}

of the evaluation of

_equation

₍₆₎

for various

_height

.

autocorrelation

_{functions (h (e ) -}

_h(0))2)

shown on the left hand side. The center column

shows the correlation function

_{C (e )}

defined in

_{equation (7)}

and on the

_right

is its Fourier

transform,

the

_intensity.

Case

_2(a)

is a

_rough

surface with a linear

_divergence

of

( (h (g) -

h

_{(0) )2)}

with

_{distance e;}

it results in a Lorenzian

_lineshape

for the cross section of the truncation rod. The Lorenzian width is

_proportional

to

[qz]2;

_the

rod is therefore

_sharp

close

_in,

but broadens

_rapidly

with

_[qz].

Fig.

2. - One-dimensional calculation of the

cross sectional

_lineshape

of a truncation rod from the

height-height

correlation function. The four cases are in the order discussed in the text. In order of

Case

_2(b)

is a flat

_surface,

_using

the definition of

_having

finite

_height

fluctuation at

_large

distance. On any distance scale there is a certain random variation of

height specified by

its

root mean _{square value}

_ho.

This results in diffraction which is

_sharp

in cross

section,

as

indicated

_by

a 8-function _{in qx.}

_However,

the

_intensity

varies with q,

according

to

This

_{Debye-Waller-like expression}

is

_analogous

to the

_description

of a «

rough

» surface

given

in reference

_[15]

in the sense that the

_intensity

_{falls away from the}

_{Bragg points}

more

_steeply

than the classical

_{1/sin2 (qz/2)}

_crystal

truncation rod. The distribution function here is Gaussian instead of

_exponential,

so the functional form of

_equation

(8)

is different from that

of reference

_[15].

In the context of

_roughening

transitions we would

_classify

this surface as

smooth because the

_height

fluctuations do not

_diverge.

The

_signature

_{is the presence of the} a-function cross section of the rod.

Case

_2(c)

is intermediate between

_(a)

and

_(b)

_{having long}

_{range flat}

_asymptotic

behaviour

and

_{linearly increasing height}

correlations at _{short range. This}

_gives

rise to both

_sharp

and broad features in the diffraction. The

_example

shown is

_{particularly simple, giving}

a

_lineshape

which is the sum of a 5-function and a Lorenzian.

The last case

_2(d)

illustrâtes the

_roughness

_{predicted by}

Chui and Weeks

_[11]

for T >»

_TR

where the

_height

fluctuations

_{diverge logarithmically}

in

_e.

This

_gives

rise to a _power

law

_lineshape

as shown. This surface is

_{asymptotically rough}

but less so than case

_2(a)

and so

has a diffraction

_lineshape

which is intermediate between case

_2(a)

and

_2(c).

Further discussion of

_roughness

models and their diffraction

_patterns

is deferred to the next section.

Equation

(6)

is a

_{simple periodic}

_{function in qz} because the

_height

function

_{h (x )}

can

_only

take discrete

_(integer)

values. If we consider the more

_general

case of a

_crystal

with a basis or one cut to _{any other} direction than

_(100),

the function assumes the

_periodicity

of the new

reciprocal

lattice instead. This is illustrated for the

_Ni(113)

surface in

_figure

lb. This

_ideally

« flat »

_{configuration}

of the surface shown is used as the _{reference xy}

_plane.

With

_respect

to the cubic lattice a

_regular

_{array of} terraces and

_steps

is

_perceived

in this

_{plane. Upon}

roughening,

the

_spacing

of the

_steps

becomes

_irregular

with

_{consequential height}

fluctuations in z. We can use the same

_{crystallographic}

coordinates for

_(x,

y,

z )

if the unit cell is defined as

shown

_by

the axes of the

_figure.

The definition is

_{orthorhombic ;}

to convert to

_physical

distance

_(À)

units

_{(x, 1)}

defined

_along

the same directions we need :

where ao is the f.c.c. lattice

parameter,

3.524 A

for Ni. The

_reciprocal

of transformation

₍₉₎

relates

_{(qx qy qz)}

to

_(qx

_qy

_qz)

in

_physical

_{(A -1)}

units.

The

_reciprocal

lattice is shown in

_figure

3.

_Crystal

truncation rods

_(CTR)

emanate from each bulk

Bragg

point

in the direction of the surface

normal,

which is 113 here. This has the consequence of

linking

the 002 reflection to

111 on

the first-order rod as shown. Since

q1 is now defined

_{by equation}

₍₉₎

in units of the

_{reciprocal layer spacing, equation}

(6)

can be

every

Bragg

point.

This

recentering

means that measurements can be made with momentum transfer

_parallel

to the surface

_(as

in the

advantageous

grazing

incidence/grazing

exit

_angle

geometry),

yet

still

_{permitting large}

values of

_[qz]

to be attained.

All measurements

_presented

here were made on the first-order CTR of

_Ni(113)

at its

_point

of intersection with the surface

_plane

_{(664 /11.}

The incidence and exit

_angles

were both

approximately equal

to the critical

_angle

for total extemal

reflection,

optimally

suppressing

background

from the bulk and

_enhancing

the

_{signal by}

refraction

_{[12, 13].}

The nearest

_Bragg

peak

is seen in

_figure

3 to be

_{(111 )}

which is a

_reciprocal

_space distance

_[qz]

= 2

7T ;1

_{away. It} 11

is therefore close to the

antinode,

[qz]

= _ar,

midway

between two

Bragg peaks

(the

« anti

Bragg » position

referred to in He

_scattering),

that

_gives

the maximum

_sensitivity

to

roughness,

as can be seen

_directly

in

_equation

_(6).

Fig.

3. -

Reciprocal

space

diagram

for the 113 cut

_through

a face centered cubic lattice. Dots

representing Bragg peaks

are indexed

_{traditionally.}

_qx

_{and qz}

are

_{respectively parallel}

and

perpendicular

to the surface as used in the text.

_Crystal

truncation rods

_(CTRs)

link

_Bragg

_{peaks together}

in the

perpendicular

direction. The

_intensity

of the rod

_depends

on the state of

_roughness

of the surface and has a functional form in

_{(qx qz)}

_{predicted by equation}

_(6).

A sensitive test of the

_{theory presented}

_{above is its qZ}

_dependence.

For

_example,

the width of the

_lineshape

_{in qx} should be

_quadratic

in the distance to the nearest

_Bragg

_point,

as should

the value of the

_{exponent q}

_appearing

in

_figure

2d and discussed below. It is

_important

to test

this behaviour because

_large

scale variations in the direction of the surface normal will also

change

the widths of the

rods,

but in a _{way that is linear in}

[qz].

For the 113 surface

_{(Fig. 3)}

the q,

dependence

is

_readily

tested because the distance of the intersection of the second order

CTR with the surface

_plane

is

_only

a distance

[qz]

= 2

1T /11

from the

(111 )

Bragg peak.

Roughening

model for T

_TR.

Here we consider the

_{terrace-step-kink}

_(TSK)

model of Villain et al.

_[4]

for the low

temperature

behaviour of a surface with a well defined « flat » state described

by

a

step-and-terrace

_picture

of

_figure

lb. The

_elementary

excitation that leads to

_roughening

is a

_pair

of

kinks

_{separated by}

a run of n

_displaced

_step

_edges.

The

_density

of such excitations in the T

_TR

limit is

_sufficiently

low that

_they

do not interact

_{significantly.}

The energy of the local excitation is

where

_Wo

_{is the kink energy and w is the energy per site} to

_displace

a

_step.

The

_height

of the surface is increased or decreased

_by

one vertical lattice

_spacing

over the n sites that are

displaced.

The mean _square

_height

is evaluated

assuming

a Boltzman distribution and

Because no interactions are assumed there are no correlations in the

_height

distribution which is therefore

_independent

_{of §.}

_{2 (h2)}

can be substituted

_{for (h (e ) -}

_{h (0) )2)}

in

_equation

₍₈₎

which leads to a &-function

_lineshape

with an

_intensity

This was the functional form used to

_{analyze X-ray}

data for

_{Cu(110) [5]}

and

_{Cu(113) [6].}

In each case the

_{integrated intensity}

was used since no

_change

in

_lineshape

was detected. The

quoted

values of

_{Wo/ w}

were 2260/170 K and 2100/65 K for

_{Cu(110) [5]}

and

_{Cu(113) [6]}

respectively.

The values of w have been

_adjusted

for

_consistency

with the definitions in

equation

(12).

Roughening

models for T>

_TR.

We will consider here the results of different theoretical

_analyses

that can be

_applied

to the

f.c.c.

₍₁₁₃₎

surface at and above its

_roughening

temperature,

TR.

We have redefined the

length

scales for

_consistency

with the definition of

_figure

lb and

_equation

_(9).

We start with the

_simple

cubic SOS model of

_figure

la. A harmonic Hamiltonian is used to describe the energy of interaction between columns

_summing

over nearest

_{neighbours ij}

This disfavours double

_height

_steps

between columns. Chui and Weeks have shown

_[11]

that the SOS model

_gives

rise to a

_roughening

transition and that the

height-height

correlation

function

_{diverges logarithmically}

with distance

_along

the surface. We will use a

_general

where the dimensionless coefficients are in

_{general temperature-dependent.}

_Cx =

Cy

= 1 for

the

_simple

cubic SOS model.

The

Body-Centered

Solid-on-Solid

_(BCSOS)

model

_[16]

is a more

_appropriate

extension of

this model to the

₍₀₀₁₎

faces of fcc metals. This model includes a sublattice of vertical columns

at

_(1/2,

_1/2,

_1/2)

times the conventional lattice vectors. This model has also been shown to exhibit a

roughening

transition

[17].

The extension of this model to the

₍₁₁₃₎

surface is

_{accomplished by applying}

the correct

periodic

boundary

in the

_parallel

direction. Kinks at

_step

_edges

are the

_major

excitation in this model since a

_kink,

unlike a _vacancy on the

(001)

terrace, has fewer nearest

_neighbor

columns with different

_heights.

Interactions between

_steps

are not

_specifically

included in this model.

However,

a _{short range}

_repulsion

is

_implied,

because

_{equation (13)}

excludes a

_significant

density

of double

_step

_heights

on the surface.

_Thus,

this effective interaction inhibits

_step

collapse.

It is worth

_noting

that the BCSOS model does not limit whether a kink is more than

one atomic row

_deep,

so that

_steps

are allowed to advance

_by

more than a

_single

row of atoms. The

_{height-height}

correlation function has been calculated

_by

Den

_{Nijs et}

al.

_[18]

to have the

same

_logarithmic

form of

_equation

(14)

for T >

_TR

with :

However,

the lack of an

_explicit

_step-step

interaction in the BCSOS model means there is no

prediction

of the

_temperature

_dependence

of X.

Another model for T>

_{TR is}

discussed

_by

Villain et al.

_[4],

_deriving

its Hamiltonian from Jose et al.

_[19],

_expressed

in the

_language

of kinks in the

_steps

_(see

_previous

_section)

instead of column

_heights.

The columns of this model are in effect

_aligned

with the terraces instead of vertical. We will also refer to it as a

_{terrace-step-kink}

_(TSK)

model.

_Explicit

step-step

interactions are included. The Hamiltonian is shown to reduce to a form similar to

equation

(13)

that retains the

_anisotropy

between

_Wo

and w when these are removed

_[4].

The

TSK model

_{again gives}

the

_logarithmic

form for the

_{height-height}

correlation function above

TR

with

_predicted

coefficients

_[4].

where w and

_Wo

are the

_energies

defined in the

previous

section,

satisfying w

« kT «

_Wo.

The model is shown to have a Kosterlitz-Thouless transition at the

roughening

_temperature

which

itself satisfies

from which the value of X at T =

TR is

seen to be the same as for the BCSOS model

equation

(6).

The

_{logarithmic dependence}

on distance

gives

rise to a

_powerlaw

correlation

function :

from which the

_intensity

is calculated

_by

Fourier transformation

where

_[qx]

and

_{[qy ]}

are also modulo 2 ir,

_being

the momentum transfer deviations from the nearest

_Bragg

rod. The

_{exponent, q,}

has the

_temperature

_dependence

of the coefficient X

At the

_{Bragg peaks,}

sin

_{(qz/2) = 0}

_{and q =}

0.

_Equation

₍₂₀₎

then

_degenerates

to the

expected

5-function form in

_{qx qy}

_{and q. From}

_equation

₍₁₈₎

or

_(15b)

it

_readily

seen that the

value of the

_exponent

at the most sensitive diffraction

_{position qz}

= 7r at the

_roughening

transition

_{is q =}

1.

Expérimental

methods.

X-ray

measurements were made on beam line X16A at the National

_{Synchrotron Light}

Source

_(NSLS)

at Brookhaven National Lab. 5 milliradians of radiation were focussed

_by

a

toroidal Pt-coated mirror at 5.67 milliradians incident

_angle.

Two

_parallel

_Si(lll)

_crystals

selected the

_wavelength

of 1.5

_Â,

chosen to lie below the Ni

_{adsorption edge.}

The

_sample

sat at the focus of the

_beam,

at the center of a 4-circle diffractometer and in a _{pressure of}

2 x

10-1°

Torr maintained

_by

_{ion pumps and} a Ti sublimator. The

_design

of our Ultra

_High

Vacuum

_(UHV)

diffractometer is

_published

elsewhere

_[20].

Resolution was determined

_by

the mirror for the incident beam and

_by

2 mm slits at 500 mm for the exit beam. The former

was 0.001

 -1 and

so was

_{unimportant ;}

the latter determined the instrumental resolution of 0.0149

 -1

radial and 0.0053

 -1

transverse at the first-order CTR

_position.

The

_sample

was cut from a

_{single crystal by spark}

erosion. The face was cut

_slightly

convex so that the exact 113 direction could be selected

_by

suitable translation. In fact this translation served a _{second purpose of}

_allowing

us to search for a

_large

flat

_region

on the

surface,

which

was

_probably

more

_important.

The surface was first

_{prepared by}

electrochemical

_etching

in 50 %

_H2SO4

at 0.5 A

cm-2,

then

_{by heating}

in

_02,

then

_{by sputtering}

and

_annealing.

Contamination of the surface

_by

S was found not to be a

_problem

below 1 150

_{K ;}

carbon contamination occurred

_{slowly during}

each

_experiment

but was never more than 1 %. The

sample

was heated

_radiatively

and the

_temperature

was measured with a chromel/alumel

thermocouple,

calibrated

_using

the thermal

_expansion

coefficient of 1.35 x

10- 5 K- 1.

The

_sample

was

_{aligned using}

the

_out-of-plane

311

and

131

bulk

_{Bragg peaks}

at 670 K to determine an orientation

matrix ;

thermal offsets at the

_roughening

_temperature

were

_thereby

reduced. Scans were made

_through

the first-order _{CTR in the qx radial}

_[parallel

to

(332)] and qy

transverse

_[parallel

to

_(110)]

directions

_extending

far

_enough

to establish a clear

background.

The

_point

on the rod was

_slightly

above

6/11 6/11

4/11 in order to achieve

Fig.

4. -

Lineshapes

measured at 6/11 6/11 4/11 with radial scans

_{(along qx}

at _temperatures above and below

_TR.

The

fitting

curves are Lorenzians convoluted with the Gaussian instrumental resolution. These 2D L * G fits were

_{performed simultaneously}

in

_qx

and

_qy.

Result and discussion T

_TR.

The

_temperature

_dependence

is

_immediately

evident in the radial

_lineshape

shown in

figure

4 : the

_{intensity drops dramatically}

in the center of the scan and rises in the

_adjacent

regions

as

_temperature

is raised. The same behaviour is seen in the

_{transverse qy}

direction. The central

_part

of the low

_temperature

_lineshape

is resolution limited in both directions. A

_{simple lineshape}

that fits

_{simultaneously}

the radial and transverse data at all

temperatures

is a

_simple

_Lorenzian,

convoluted with the known resolution

_parameters,

assuming

this to be Gaussian. We denote these L * G fits. Goodness of fit was established

_by

the conventional

_{X 2}

_{considering only counting}

statistics as the source of

_experimental

error.

For these fits a

_height,

two

_widths,

two

_peak

_positions

and a

_background

were the

_parameters

and the result was a

X2

of 2.7

_averaged

over all

_{temperatures,}

will the worst case

2= 3.6.

Figure

5a shows the

_dependence

of

_{peak height}

with

_temperature.

It follows

_roughly

the

same TSK curve from the Villain

_theory

_{[4] (Eq. (12))}

that

_explained

the

_Cu(113)

data

_[6]

with a

_roughening

_{temperature -}

150 K

_higher.

However closer

_inspection

shows this curve to be

_inadequate

at the

_high

T end : the full and dashed TSK curves shown have values of

Wo/ cv

of 4 300 K/10 K and 2 700 K/100 K which are both at the limits of

_{acceptability}

for T 720 K. Above 720 K the

_agreement

is bad. From

_{equation (17)}

we conclude that

TR

= 740 ± 20

K,

assuming

the two _cases shown to be the _extrema of the fit.

Wo

= 4 000 K is

already

of the order of the estimate _{of the energy of the} Ni-Ni bond that would be broken to form the kink if there were no relaxation

_[4],

so

_higher

values would be considered

unreasonable.

Note that

_Debye-Waller

corrections have not been included in the

_{analysis simply}

because of the small momentum transfers used in these

_experiments,

_{1 Q 1 = 1.5 Â -1.}

These

conditions in these

_experiments

_[21].

Values from He

_scattering

measurements on

_Ni(113)

would

_{similarly give}

2 M = 0.06

[22].

The

_exponent

in

equation

(12)

used to fit the data in

figure

5,

however,

ranges between 3.3 to 9.4 so that

_Debye-Waller

correction to the fits are

negligible.

Corrections to

_Wo

and w from inelastic

_scattering

are much less than 1 % and can

be

_ignored.

The low value of 2 M has

_important

_{consequences in the}

_analysis

of diffraction

line

_shapes

as discussed in the next section.

Fig.

5. -

Temperature dependence

of the

_{peak height}

of the first order CTR at 6/11 6/11 4/11 obtained

by

means of L * G fits.

(a)

Theoretical curves are from the T «

_TR

limit of the TSK model

_{(Eq. (12)}

and Ref.

_[4])

with _parameter values

_Wolw

indicated. The dashed curve marked

_Cu(113)

is the best fit to the behaviour of that surface measured

_by

_{Liang et}

al.

_{(Ref. [6])}

with

_Wo

= 2 100 K w = 65 K.

(b)

Theoretical curves for the T =5

_TR

_{renormalization group}

_expansion

of the TSK model

_{(Eq. (24)}

and Ref.

_[4]),

with _parameter values

_Wo

and

TR

indicated.

The

_explanation

we favour for the

_disagreement

in

_figure

5a is that we are

_seeing

the breakdown of the Villain T «

_{TR theory}

in the

_vicinity

of

_TR.

This means the values of

Wo

and w obtained here and in

_previous

work should have much

_larger

error bars than

claimed

_[6],

as

_figure

5 demonstrates.

_However,

it is still hard to account for the

_discrepancy

in these fitted values with He

_scattering

values for

_Wo

and w

_[23].

The breakdown of the TSK T «

_TR

_theory

is not a

_surprise

when T -

_TR

because the

_density

of kinks is so

_{high they}

can no

_longer

be considered isolated and

_{non-interacting.}

Villain et al.

_[4]

have in fact derived another

_expression

for the

_height

correlation function

renormalization

_description

of the TSK model mentioned

above,

that

_predicts

Kosterlitz-Thouless behaviour at the transition :

where

_K(§)

_{decays exponentially}

at

_{large e}

and a is a constant that

_depends

on

WO,

Neglecting

the

_finite-range

_C-dependent

term which

_gives

rise to a broad

_component

in the

diffraction,

as it does in

_figure

2c,

we can substitute

_equation

₍₂₂₎

into

_equation

₍₈₎

to obtain the

_temperature

_dependence

of the

_{sharp, Bragg}

_{component :}

Figure

5b shows fits to

_equation

₍₂₄₎

for different values of

_Wo

and

_TR.

The best overall fit is with

_Wo

= 1 500

K,

TR

= 760

K ;

values of

Wo

above 3 000 K _are ruled _out, even

_though

it is rather insensitive to this

_parameter.

Between the two

_limiting

cases of the TSK

_{theory, equation}

₍₁₂₎

and

_equation

_(24),

we therefore cover _{the whole range up} to

_TR

_using

more or less the same

_parameter

values. The

non-interacting

kink

_{approximation}

works for r

_TR

while the renormalization _group

expansion

works for T ,

_TR

as

_expected.

The

_tailing

of the data

_just

above

_TR

could not be

explained

in the TSK

_model,

but can be understood with the KT

_description

as either the

height

of the diffuse

_component

above

_TR

or

_{inhomogeneous broadening}

of the transition due

to strain

_gradients

in the

_sample.

The appearance of a broad

_component

_{starting just}

below

_{TR is}

confirmed in

_figure

6

by

the

T

_dependence

of the Lorenz widths from the same fits. The widths are constant until a

threshold value is reached near T = 740

K,

and

diverge

above. The threshold is not

sharp

as it

is for other surface

_phase

transitions

_[14],

so it is not

_possible

to determine an accurate transition

_temperature

_{in this way. The difference between the widths in the} transverse and radial direction is a measure of the surface

_anisotropy

defined in

_equations

(14)

and

(16a)

which is discussed below.

Results and discussion T:>

_TR.

To check for

_consistency

with the

_prediction

of

_{logarithmic divergence}

of the

_{height-height}

correlation function for T>

_{TR and}

to

_investigate

the

_uniqueness

of the

_fitting

_methods,

we

have also fit the diffraction

_profiles

in two _{further ways.}

_First,

a one dimensional convolution

of

_equation

₍₂₀₎

was made with the Gaussian instrument response function. The resolution

function,

which includes finite size

effects,

was determined from the bulk

_{peak profiles}

as before. The

_adjustable

_parameters

in the

_power-law

fits were the

_exponent

n, the

peak

height,

and

_background.

The

_quality

of the

_power-law

fits was

_compared

with the

L * G fits over the entire

_experimental

temperature

range. Below 750 K the

X 2 values

of L * G were about a factor of 2 lower than the power law fits. Above 750 K both

_fitting

Fig.

6.

_{- Temperature dependence}

of the radial

_(qx)

and transverse

_(ily)

Lorenzians widths obtained from two dimensional L * G fits

_including

those shown in

figure

4.

Fig.

7. -

Typical

radial scan across the CTR

_connecting

the

₍₀₀₂₎

and

_(III)

bulk reflections. The

intensity

is

_plotted

as a function of the momentum transfer

_qx.

The scan was taken at a value of

[qz]

= 0.91

7r, close to the

_out-of-phase

condition. The solid line is a fit to the data

_using

the power law form of

_equation

₍₁₇₎

convoluted with the finite

_crystal

size as

_explained

in the text.

While q

is

_strictly

zero below

_{TR, fitting}

the data to a _{power law}

_{peak shape}

over the entire

temperature

range is instructive for two reasons.

_First,

it allows a

_view,

_by

_comparison

with

the L * G

fits,

of the evolution of the

_{lineshape through}

the transition.

_Second,

the value

of q

below

_{TR is}

a measure of both the

_{one-phonon broadening}

and any residual defect

_broadening

of the low

_temperature

_{peak shapes}

_{[9, 22].}

The latter

_point

is

_particularly

useful in

_judging

the accuracy of the

lineshape

fits due to corrections from these effects.

A second and more intuitive method to obtain the

_{exponent 17}

is to

_plot

the data

logarithmically

as in

_figure

8. From

_equation

_(17),

the

_slope

of the

_{log-log plots}

is

proportional

to 11. The

_advantage

of this method is that data

_points

near the

_peak,

which are

Fig.

8.

_{Log (I )}

versus

_log(q)

for radial

_qx(A)

and transverse

_qy(e)

directions taken at _temperatures above and below

TR.

At a

_given

_temperature the linear

_region

of the

_{log-log plots}

for both radial and transverse directions have the same

_{slope of q -2.}

The _asymmetry

_{between 4F,}

_{and qy}

is

_clearly

evident as a horizontal shift.

ignored :

all of the fits excluded data less than 2 0- _{away from the central}

_peak.

This means

that q

was determined over a _{range of correlation}

_lengths

between

1300 Â

to 100

_Â,

_probing

ten times further than

_previous

studies

_[7-9].

The values of ry determined

_by

_{the power law}

peak profiles

and those measured from the

_{log-log plots agreed}

with each other within the

error bars of the

_{X 2}

_fitting

routines. This

_justifies

our claim that instrumental and finite-size effects are

_unimportant

in the full

_lineshape

fits.

At each

_temperature

the values

_{of q}

determined

_separately

from the radial and transverse

scans were found to be

_essentially

the same, as

_figure

9 shows.

_Using

_{equation (21),}

the

roughening

temperature

was determined from the

_temperature

at which

X(T)

=

1/ ’TT 2 :

using

the value of 10

_ir/11

_{for qz, ’TJ should have} a value of 0.83 at

_TR.

From

_figure

9 this

_gives

a

roughening

temperature

of 770 K ± 30 K. This value is in

_good

_agreement

with that obtained

in

_previous

atom

_scattering

studies of

_Ni(113)

_{analyzed by}

the same method

[8].

It also agrees

with the value in the

_previous

section for the

_intensity

_dependence

at T

_TR.

The

_temperature

_dependence

of 17 is also

predicted

for the TSK model

through

equations

(16)

and

_(21).

Drawn in

_figure

9 are two curves with different values of the TSK

parameters.

The curve

_{corresponding}

to the T

_TR

_parameters

of

_figure

5

_gives

too small a

slope

for q

(T)

near T =

TR

_to

explain

the data. It is necessary to increase

Wo

to 104 K in

order to

_get

a

_good

_{fit ;}

this value is

_totally

_unphysical

however

[4].

Our data are

_{significantly}

different from the atom

_scattering

measurements for

_{Ni(113) [8, 17], Cu(113) [23]}

and

Cu(115) [9, 24]

which

find q

oc T over a wide range of T and see

_only

a small

_change

in

_slope

of q

vs. T near

_TR.

The reason for this is the much smaller

_{one-phonon braodening}

in the

X-ray

experiments.

In atom

scattering experiments,

the total momentum transfers are

_large,

typically

5.9

Å -1

_{[7-9, 24],}

and

the n

(T)

curve is

_{significantly}

affected

_by

the way the

_fitting

Fig.

9. - Plot of the

exponent, 71, versus _{temperature. Data for both qx and qy} directions are included. The

_roughening

_temperature is determined to be 770 ± 30 K from the _temperature at which

X ( T )

equals

1/1T2,

hence q

= 0.83. The dashed _curve in from

equations

(16b)

and

(21)

corresponding

to the TSK model with

_Wo

= 3 500 K w = 32 K that _was used in

figure

5. The full curve has

Wo

=

104 K w

= 0.007 K.

of reference

_[18]

shows this

_clearly.

In our

_X-ray

_{measurements,} with a total momentum transfer of 1.5

_Â-1,

the number of

_{one-phonon scattering}

events

_(those

_{primarily responsible}

for distortions of the

_{lineshape), proportional}

_to

_{1 Q}

₂

_[26],

is ten times smaller. Inelastic

scattering probably

does account for the non-zero effective value

of 77 -

0.4 seen for T «

_TR.

The TSK

_{theory predicts}

17 =

0,

whereas pure

one-phonon

inelastic

scattering

would

give

a

_lineshape

_{with q =}

1. Without correction the atom

_{scattering experiments}

_{gave values}

of 17 ’"

0.6 to 0.8

_{[7-9, 24]}

for T «

_TR.

We therefore

expect

our determination

_{of q}

to be more

reliable.

Because both the TSK and BCSOS models for the

_roughening

transition

_presented

so far

predict

a

_{power-law lineshape}

for T>

_TR,

a more critical test, which differentiates between the

theories,

is to examine the

_lineshape

_asymmetry,

which is

_directly

related to the surface

tension

_{anisotropy. X-ray scattering}

is a

_{particularly important technique}

here because of its

high

resolution.

The

_peak

_asymmetry

is characterizd

_by

the

parameter

_cx/cy

in

_{equations (14)}

and

_(20),

which,

for the TSK model as

_equation

_(16a)

_indicates,

is a relative measure of the ratio of the kink energy,

Wo,

to the

_step-step

_{interaction energy,} co. We have measured this

_parameter

in

three ways.

First,

a full 2D convolution of the instrument

_(plus

finite size

_effects)

with the

power-law

form of

_equations

_(17),

and

₍₁₈₎

was

_performed.

The second method uses the

observation

_that,

if the radial and transverse scans are

_plotted

on the same

_logarithmic

scale

(Fig. 8),

the

_asymmetry

is the

_displacement

_{in q}

between the curves. As in the determination

of the

_exponent

n, this method reduces errors associated with the detector convolution. These

two methods _{gave the} same values of

cx/cy

within error. The third method is to take the ratio

of the widths of the L * G fits from

_figure

6. This gave

_slightly

different results. We have

retained the normalized

_system

of

_{(qx qy )}

units of

_figure

1b. In natural

_{(qx qy)}

units the

asymmetry

is

V11/4

times less

pronounced

according

to

_{equation (9).}

The results are

_plotted

in

_figure

10.

In these

_experiments

_cx/cy

is found to increase with

_temperature.

From

_equation

_(16a)

the TSK model

_predicts

a decrease with

_temperature

above

_TR

since kT

_Wo.

Atom

scattering

experiments

found values that were constant over a wide range of

_{temperature ;}

the

Fig.

10. - The

_lineshape

_asymmetry

_cx/cy

as a function of _temperature. This reflects the

_anisotropy

of the surface tension. Fits to the data of both

_powerlaw

_(e)

and L * G

_{(A) lineshapes}

were used and differ

slightly.

The theoretical curves refer to the TSK model discussed in the text. _{The range of parameters} shaded for the TSK model is 2 300

_Wo

2 600 K constrained so that

_TR

= 740 K

_by

_{equation (17).}

Although

we find

_{large discrepancies}

for T

_TR

where the

_theory

is not _{valid anyway, the}

region

T:>

_TR

shows a

_relatively

flat

_temperature

_dependence

that can be considered consistent with

_theory.

The numerical value of

_cx/cy

here of 8.5 ± 1

_requires

Wo

= 2 450 K _± 150 K when

TR is

fixed at 740 K

_{(not critical)}

as shown. The value of

Wo

= 3 500 K found for

T « TR

(Fig. 5)

would

_predict

_cx/cy

= 34.

The BCSOS model

_{(Eq. (15a))}

_predicts

_cx/cy

= 1 which is

clearly

invalid. The

_asymmetry

we observe lies between the

_predictions

for the two models. We can

_suggest

therefore that an

intermediate

model,

for

_example

the

_anisotropic

BCSOS model in which the

_anisotropy

is

adjustable

[16],

might

be better.

Summary

and conclusions.

We have

_presented

a

_theory

which shows how

_X-ray

diffraction

profiles

of

crystal

truncation

rods

_(CTR)

can be

_interpreted

in terms of statistical-mechanical models of

_roughening.

We used this to

_analyse

data for the lst order CTR of

_Ni(113)

as a function of

_temperature.

A

simple

Lorenzian

_lineshape

was found to be

_{indistinguishable}

from the

_expected

_power-law

lineshape

in the range T >

TR,

but gave a better fit for T

_TR.

This lack of

_sensitivity

shows that the data are still

_inadequately

accurate to _prove the correctness of the

_logarithmic

divergence

of

_height

with distance at T >

_{TR predicted by}

Chui and Weeks

_[11],

_{although they}

are

_certainly

consistent.

Below

_TR

we find the

_lineshape

to be a resolution-limited 2D

_{Bragg peak,}

whose

_intensity

dependence (Fig. 5)

on T for T «

_{TR is}

consistent with the

_{non-interacting}

kink

_theory

of

Villain et al.

_[4].

From this we obtain the

_microscopic

_parameters

of the TSK model :

Wo

= 3 500 ± 800

K, co

= 55 ± 45 K consistent with a value of the

_roughening

_temperature

TR

= 740 ± 20 K. We see

significant

deviation from the

theory

as

_{T approaches}

_TR,

and much better

_agreement

with the renormalization _group

_analysis

of the same model

_{[4]. Consistency}

with the

_prediction

of Kosterlitz-Thouless behaviour near

_{TR is}

observed with reasonable

parameter

values 500 K

_Wo

3 000 K and

_TR

= 760 ± 10 K.

Above

_TR

we find

_significant

disagreement

with the TSK model.

Firstly,

for any reasonable

values of the

_microscopic

_parameters,

our

_experiment

shows a much

_{steeper temperature}

dependence

(Fig. 9)

of the

_lineshape

exponent q

near

_TR

than

_predicted.

This

_steep

_slope

also

contributions to the

_lineshape.

The determination of

_TR

= 770 ± 30 K from the

crossing

of

7y

(T)

through unity

appears to be reliable and consistent with the T

_TR

result.

_Secondly,

we measure a

_{lineshape anisotropy}

_{(Fig. 10)}

of

_cx/cy

= 8.5 _± 1 for T >

TR, compared

with 1.25 for

_{Ni(113) [8],}

2.0 for

_{Cu(113) [23]}

and 5.0 for

_{Cu(115) [24]}

measured with atom

_scattering,

and

_compared

with 34

_required

from the TSK

_theory

with the above

_parameter

values.

Lowering Wo

to 2 450 ± 150 K is needed for

_consistency

here

_{(Eq. (16a)).}

The alternative

BCSOS model

_[18]

_predicts

no

_anisotropy

and so can be ruled out,

although anisotropic

versions

_[16]

could be viable.

We note in

_closing

that the T

_{TR and}

T >

_TR

TSK

_parameters

are about an error-bar

apart.

The various

_parameter

determinations are listed in table 1. A consensus set of

parameters

marginally

consistent with all data

_{(except n (T))}

would be

_Wo

= 2 650 ± 300

K,

w = 110 ± 80 K

giving

TR

= 755 ± 15 K. This would appear to scale well with the

X-ray

results for

_{Cu(113) [6] [Wo}

= 2 100 K = 65

K ],

but not with the

atom-scattering

values for

Ni(113) [8] [Wo

= 1 700 ± 500

K, w

= 200 ±

_{100 K ]}

or for

Cu(113) [23] [Wo

= 800 ±

80

K, w

= 560 _± 60

K ],

although

this may be because of different

analysis

techniques.

The

main new

_finding

of this work is the

_large

value of the

_anisotropy

_cx/cy

above

TR.

This is

_directly

_apparent

from the raw linewidth data

(Fig. 6),

but also

clearly

seen in the

lineshape analysis.

Table I. -

Summary

of

fit

parameters

for

the TSK model

_{(Ref. [4])}

derived

_{from different}

analyses of

the

_X-ray

data. Measured values

(in

Kelvin)

are

stated,

while those derived

from

equation

(17)

are in

parentheses. Theory

A

refers

to the

_{non-interacting}

kink

_{approximation}

(Eqs. (10-12)

and

_Ref.

_{[4]) ;}

_theory

B

_refers

to the renormalization _group treatment

_of

reference

[4].

Acknowledgements.

We wish to thank J. D.

Axe,

J. Bohr and S. G. J. Mochrie for

_helpful

discussion on the

theory

of diffraction from

_rough

surfaces and V.

_Straughan

of

_Monocrystals

Inc. for

_cutting

the

_crystal.

NSLS is

_{supported by}

DOE Grant No. DE-AC012-76CH00016. Work at U. of Missouri was

_{supported by}

the NSF under Grant No. DMR8703750 and the U. of Missouri

Faculty Development

Fund.

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