Unit 4

The Laue equations

• Let a, b, c be the primitive vectors of the crystal lattice L

whose atoms are located at the points
x=pa+qb+rc
that are integer linear combinations of the primitive vectors.
Let kin be the wave vector of the incoming (incident) beam,
and let kout be the wave vector of the outgoing (diffracted)
beam.
Then the vector kout - kin = k is called the scattering
vector (also called transferred wavevector) and measures
the change between the two wavevectors
• The three conditions that the scattering vector k
must satisfy, called the Laue equations, are the
following: the numbers h,k,l determined by the
equations

Hence there are infinitely many scattering vectors that satisfy the
Laue equations. This condition allows a single incident beam to be
diffracted in infinitely many directions. However, the beams that
correspond to high Miller indices are very weak and can't be
observed. These equations are enough to find a basis of the
reciprocal lattice, from which the crystal lattice can be determined.
This is the principle of x-ray crystallography.
 It’s a single crystal
200

220

111 222
311

2q

At 27.42 °2q, Bragg’s law The (200) planes would diffract at 31.82 The (222) planes are parallel to the (111)
fulfilled for the (111) planes, °2q; however, they are not properly planes.
producing a diffraction peak. aligned to produce a diffraction peak
 A random polycrystalline sample that contains thousands of
crystallites should exhibit all possible diffraction peaks

200

220

111 222
311

2q 2q 2q

• For every set of planes, there will be a small percentage of crystallites that are properly
oriented to diffract (the plane perpendicular bisects the incident and diffracted beams).
• Basic assumptions of powder diffraction are that for every set of planes there is an equal
number of crystallites that will diffract and that there is a statistically relevant number of
crystallites, not just one or two.
 Powder Samples
200
NaCl • Salt Sprinkled on double
<100>
Hint stick tape
Typical Shape
Of Crystals
• What has Changed?
111
220 311 222

It’s the same sample sprinkled on double

stick tape but after sliding a glass slide
across the sample
 Integral breadth method
• Integral breadth (IB) methods proposed by Klug & Alexander, 1974
are frequently used in studies of the microstructure of material for
a quick estimation of the so-called ‘size–strain’ line broadening
effect, mostly relating to the broadening caused by the average size
of the crystallites (coherently scattering domains) and by lattice
strains (often denoted as microstrains or lattice distortions) caused
by, e.g. the presence of lattice defects.
• Traditional IB analysis is based on the well known Scherrer
formula (Scherrer, 1918) or on the Williamson–Hall (WH)
method (Williamson & Hall, 1953)
• IB methods are used best when trying to identify trends in the
change of the microstructure by comparing specimens belonging to
a series of experiments.
• The Scherrer formula, stating that the integral
breadth β(d*) is inversely proportional to the
apparent size only addresses line broadening
that is entirely caused by size effects.
• The Scherrer Equation was published in 1918
• Peak width (B) is inversely proportional to
crystallite size (L)
X-Ray Peak Broadening is caused by
deviation from the ideal crystalline lattice
The Laue Equations describe the intensity of a diffracted
peak from an ideal parallelopipeden crystal
• The ideal crystal is an infinitely large and perfectly
ordered crystalline array
– From the perspective of X-rays, “infinitely” large is a few
microns
• Deviations from the ideal create peak broadening
– A nanocrystallite is not “infinitely” large
– Non-perfect ordering of the crystalline array can include
• Defects
• Non-uniform interplanar spacing
• disorder
<
• Many factors may contribute to
the observed peak profile
• Instrumental Peak Profile
• Crystallite Size
• Microstrain
– Non-uniform Lattice Distortions
– Faulting
– Dislocations
– Antiphase Domain Boundaries
– Grain Surface Relaxation
• Solid Solution Inhomogeneity
• Temperature Factors
• The peak profile is a convolution of the profiles from
all of these contributions
• In summary, critical points to be considered in evaluating
the reliability and quantitative meaning of WH results are as
follows.
(a) Real polycrystalline materials are most frequently
polydisperse systems, i.e. crystallites have different sizes and
often different shapes. This affects widths and shapes of all
diffraction line profiles. For simple polyhedra, size
distribution and shape can be decoupled in principle , but
this separation is impossible using the integral breadths only.
• At most, crystal shape can be considered by using the
appropriate Scherrer constants to model the size-induced
anisotropic line broadening [different integral breadths for
the various crystallographic directions. Further, line profiles
produced by dispersed small crystallites (e.g. according to a
lognormal or to a Gamma distribution) are non-Voigtian and,
as a consequence, equations (1) and (2) are not appropriate.
• The restrictions discussed above are also implicitly
assumed by the WH method, which essentially combines
the Scherrer equation with an expression for the
apparent strain obtained by differentiating Bragg’s law
The most popular expressions used for a WH analysis are

and

• where (d*) is the total integral breadth, S and D are,

respectively, the integral breadth components for size
and strain, and e is as an upper limit for the lattice strain.
• Once the instrumental component has been
removed from the observed profiles and
integral breadth values extracted from the
data, a simple plot (usually named the WH
plot) of versus d* (or β2 versus d*2) provide
estimated values for <Li>v and e from the
intercept and slope, respectively
 Williamson-Hall Plot
K 
FWHM cos    Strain 4 sin  
Size

y-intercept slope
(FWHMobs-FWHMinst)
cos(q )

nin g
d e
b roa
i n
s tra
d
an
iz e
ns
G ra i
Grain size broadening

4 x sin( ) K≈0.94
Gausian Peak Shape Assumed
• Williamson-Hall (W-H) analysis is a simplified integral
breadth method where both size-induced and strain-
induced broadening are deconvoluted by considering
the peak width as a function of 2θ.
• The significance of the broadening of peaks
evidences grain refinement along with the large
strain associated with the powder.
• The average nanocrystalline size calculate by using
Debye-Scherrer’s formula
• The strain induced in powders due to crystal
imperfection and distortion was calculated
using the formula:

From Equations 1 and 2, it is confirmed that the peak

width from crystallite size varies as 1/ cosθ strain varies as
tanθ.
Assuming that the particle size and strain contributions to
line broadening are independent to each other and both
have a Cauchy-like profile, the observed line breadth is
simply the sum of Equations 1 and 2
The above equations are W-H equations. A plot is drawn
with 4sinθ along the x-axis and βhkl cosθ along the y-axis.
The crystalline size was estimated from the y-intercept,
and the strain ε, from the slope of the fit
Warren-Averbach’s Fourier method
• In the last five decades, defect structure analysis of
polycrystalline materials by X-ray powder diffraction technique
has been able to cover a number of lattice imperfections. They
are often called size-strain analyses and usually include the
evaluation of average crystallite sizes, crystallite size
distributions, microstrains, dislocation densities and stacking
fault probabilities. They are generally based on two different
approaches1-3 and have been incorporating an impressively high
number of modifications and nuances that is to hard too mention
all of them4-10. Both approaches require a twofold task: the
determination of the intrinsic (pure) physical diffraction line
profile of the sample and, then, the extraction of the lattice
imperfection parameters.
• The first highly elaborated approach of size/strain
analysis is Warren-Averbach method (known also as a
Warren-Averbach-Bertaut method) which employs the
deconvolution Fourier-transform method (known also
as the Stokes method) for the determination of the
intrinsic physical line profile, followed by the Fourier
method for evaluation of lattice imperfections.
• This method states that the Fourier coefficients for
intrinsic physical line profile are the product of two
terms: the size and the strain coefficient. The
coefficients are numerically calculated. The method
was initially proposed for application in the field of
metals and alloys12-15, but rapidly was extend to the
area of ceramic16 and also polymer materials
• The dynamical diffraction theory of perfect crystal predicts
existence of very sharp diffraction lines with some small
inherent breadth, which results of the uncertainty principle.
• Each diffraction line is also broadened, unavoidably, by spectral
dispersion and instrumental factors. However, every
polycrystalline material with microstructure consisting of grains
less than 1 mm and some lattice imperfections (any type of
point, line or planar defects) will show some additional
broadening effect. The part of the diffraction line that is the
consequence of the sample microstructure is called, most
frequently, physical diffraction line. Therefore, the shape and
breadth of the experimental diffraction line are the mixture of
different effects.
• For the evaluation of microstructural (defect structure)
parameters of interest in materials science and metallurgy, such
as, average crystallite sizes, crystallite size distributions,
microstrains and some others, it is necessary to deal only with
physical part of diffraction line.
• The convolution theorem can be employed to disentangle the
specimen-related broadening contributions described by f(s).
• In fact, let us suppose, as in the Williamson–Hall method, that size
and microstrain are the only two sources of specimen-related
broadening.
• We call the Fourier transform of the profiles broadened by size and
distortion effects only and , respectively. As the size and distortion
profiles are folded into f(s), the following holds:

• The separation of the size and distortion terms is straightforward for

spherical domains: the size effects for a sphere are independent of
the reflection order, whereas those related to distortions (causing the
change in the slope of the Williamson–Hall plot) are order-dependent.
To describe the distortion term it is convenient to follow the idea of
Bertaut considering the specimen as made of columns of cells along
the c direction.
• The profile due to distortions is calculated by taking
the average phase shift along the column due to the
presence of defects. The analytical formula for the
distortion term is thus of the type = 〈exp(2πiLnɛL)〉,
where ɛL = ΔL/L is the average strain along c calculated
for a correlation distance (i.e. Fourier length) L.
• As a first-order approximation, the distortion terms
give no profile asymmetry; is just a cosine Fourier
transform. We can thus expand it as
• Equation represents a line in the variable n2: the intercepts at
increasing L values provide the logarithm of the Fourier size term,
whereas the slopes give the microstrain directly (Warren, 1990). From the
size coefficients, we can obtain an average size, again following the idea
of Bertaut , related to the properties of the Fourier transform:

• This average size is thus related to the initial slope of the Fourier
coefficients [assuming that they are well behaved, i.e. that the tangent is
always below the curve].
• A long chain of operations is needed to obtain the size and the strain
contributions; there is thus a risk that the final result will no longer be
compatible with the experimental data. Only a few years after the
introduction of this method, Garrod et al. (1954) wrote
• Hence, in any attempt to distinguish between particle size or strain
broadening from a particular material, the use of the one or the other of
these functions [Gaussian and Lorentzian] (together with the appropriate
relationship between B, b, and β) involves an intrinsic initial assumption
about the cause of the broadening, when the object of the investigation is
to discover the cause.
• Such an assumption must inevitably weight the experimental results,
partially at least, in favour of one or the other of the two effects. In this
connation it is therefore perhaps significant that in most previous work on
the cause of line broadening from cold-worked metals, those investigators
who have used the Warren relationship between B, b, and β have
concluded that lattice distortion was the predominant factor, whilst those
who have employed the Scherrer correction found that particle size was
the main cause. The best procedure in such work therefore is to make no
assumptions at all about the shape of the experimental line profiles…
• This is owing to the fact that a direct connection between the
experimental data and the final microstructural result does not really exist
in those methods and that the whole information contained in the pattern
is not exploited.
 Scanning tunneling electron microscope
(STEM)
• Scanning tunneling electron microscope (STEM), type of microscope
whose principle of operation is based on the quantum mechanical
phenomenon known as tunneling, in which the wavelike properties
of electrons permit them to “tunnel” beyond the surface of a solid
into regions of space that are forbidden to them under the rules of
classical physics.
• The probability of finding such tunneling electrons decreases
exponentially as the distance from the surface increases.
• The STEM makes use of this extreme sensitivity to distance. The sharp
tip of a tungsten needle is positioned a few angstroms from the
sample surface. A small voltage is applied between the probe tip and
the surface, causing electrons to tunnel across the gap. As the probe
is scanned over the surface, it registers variations in the tunneling
current, and this information can be processed to provide a
topographical image of the surface.
• The STM appeared in 1981, when Swiss physicists Gerd
Binnig and Heinrich Rohrer set out to build a tool for
studying the local conductivity of surfaces. Binnig and
Rohrer chose the surface of gold for their first image.
When the image was displayed on the screen of a
television monitor, they saw rows of precisely spaced
atoms and observed broad terraces separated by steps
one atom in height.
• Binnig and Rohrer had discovered in the STM a simple
method for creating a direct image of the atomic
structure of surfaces. Their discovery opened a new era
for surface science, and their impressive achievement
was recognized with the award of the Nobel Prize for
Physics in 1986.
The components of a STM
• Scanning tip:
Electrons tunnel from the scanning tip to the sample, creating the
tunnelling current.
• Piezoelectric controlled scanner:
Piezoelectric crystals expand and contract very slightly depending on the
voltage applied to them and this principle is used to control the
horizontal position x, y, and the height z of the scanning tip.
• Distance control and scanning unit:
Position control using piezoelectric means is extremely fine, so a coarse
control is needed to position the tip close enough to the sample before
the piezoelectric control can take over.
• Vibration isolation system:
STM deals with extremely fine position measurements so the isolation of
any vibrations is very important.
• Computer:
The computer records the tunneling current and controls the voltage to
the piezoelectric tubes to produce a 3-dimensional map of the sample
 Procedure
• The tip is brought close to the sample by a coarse positioning
mechanism that is usually monitored visually. At close range, fine
control of the tip position with respect to the sample surface is
achieved by piezoelectric scanner tubes whose length can be
altered by a control voltage.
• A bias voltage is applied between the sample and the tip, and the
scanner is gradually elongated until the tip starts receiving the
tunneling current. The tip–sample separation w is then kept
somewhere in the 4–7 Å (0.4–0.7 nm) range, slightly above the
height where the tip would experience repulsive interaction
(w<3Å), but still in the region where attractive interaction exists
(3<w<10Å).
• The tunneling current, being in the sub-nanoampere range, is
amplified as close to the scanner as possible. Once tunneling is
established, the sample bias and tip position with respect to the
sample are varied according to the requirements of the
experiment.
• As the tip is moved across the surface in a discrete x–y
matrix, the changes in surface height and population of the
electronic states cause changes in the tunneling current.
Digital images of the surface are formed in one of the two
ways: in the constant height mode changes of the
tunneling current are mapped directly, while in
the constant current mode the voltage that controls the
height (z) of the tip is recorded while the tunneling current
is kept at a predetermined level.
• In constant current mode, feedback electronics adjust the
height by a voltage to the piezoelectric height control
mechanism. If at some point the tunneling current is below
the set level, the tip is moved towards the sample, and vice
versa. This mode is relatively slow as the electronics need
to check the tunneling current and adjust the height in a
feedback loop at each measured point of the surface.
• When the surface is atomically flat, the voltage applied to the
z-scanner will mainly reflect variations in local charge density.
But when an atomic step is encountered, or when the surface
is buckled due to reconstruction, the height of the scanner will
also have to change because of the overall topography.
• The image formed of the z-scanner voltages that were needed
to keep the tunneling current constant as the tip scanned the
surface will thus contain both topographical and electron
density data. In some cases it may not be clear whether height
changes came as a result of one or the other.
• In constant height mode, the z-scanner voltage is kept
constant as the scanner swings back and forth across the
surface and the tunneling current, exponentially dependent
on the distance, is mapped. This mode of operation is faster,
but on rough surfaces, where there may be large adsorbed
molecules present, or ridges and groves, the tip will be in
danger of crashing.
• The raster scan of the tip is anything from a 128×128 to a
1024×1024 (or more) matrix, and for each point of the
raster a single value is obtained. The images produced by
STM are therefore grayscale, and color is only added in
post-processing in order to visually emphasize important
features.
• In addition to scanning across the sample, information on
the electronic structure at a given location in the sample
can be obtained by sweeping the bias voltage (along with
a small AC modulation to directly measure the derivative)
and measuring current change at a specific location.
• This type of measurement is called scanning tunneling
spectroscopy (STS) and typically results in a plot of the
local density of states as a function of the electrons'
energy within the sample.
• The advantage of STM over other
measurements of the density of states lies in
its ability to make extremely local
measurements.
• This is how, for example, the density of states
at an impurity site can be compared to the
density of states around the impurity and
elsewhere on the surface.
• Operating Principles
• The STM is an electron microscope with a resolution sufficient
to resolve single atoms. The sharp tip in the STM is similar to
that in the scanning electron microscope (SEM), but the
differences in the two instruments are profound.
• In the SEM, electrons are extracted from the tip with a series
of positively charged plates placed a few centimetres
downstream from the tip. The electrons at the apex of the tip
are confined to the region within the metal by a potential
barrier. The attractive force from the positive charge on the
plates is sufficient to permit the electrons to overcome the
barrier and enter the vacuum as free particles. The apertures
in the downstream plates form an electron lens that converts
the diverging beam from the tip into a beam converging to a
focus on the surface of the sample.
• In the STM, the plates that form the lens in the
SEM are removed, and the tip is positioned close
to the sample. The electrons move through the
barrier in a way that is similar to the motion of
electrons in a metal.
• In metals, electrons appear to be freely moving
particles, but this is illusory.
• In reality, the electrons move from atom to atom
by tunneling through the potential barrier
between two atomic sites. In this circumstance,
the tunneling electron can move either to the
adjacent atoms in the lattice or to the atom on
the tip of the probe.
• In a typical case, with the atoms spaced five angstroms
apart, there is a finite probability that the electron will
penetrate the barrier and move to the adjacent atom.
The electrons are in motion around the nucleus, and they
approach the barrier with a frequency of 10 17 per second.
• For each approach to the barrier, the probability of
tunneling is 10−4, and the electrons cross the barrier at
the rate of 1013 per second. This high rate of transfer
means that the motion is essentially continuous and
tunneling can be ignored in metals.
• Tunneling cannot be ignored in the STM; indeed, it is all-
important. When the tip is moved close to the sample,
the spacing between the tip and the surface is reduced to
a value comparable to the spacing between neighbouring
atoms in the lattice.
• The tunneling current to the tip measures the density of
electrons at the surface of the sample, and this
information is displayed in the image. In semiconductors,
such as silicon, the electron density reaches a maximum
near the atomic sites.
• The density maxima appear as bright spots in the image,
and these define the spatial distribution of atoms.
• In metals, on the other hand, the electronic charge is
uniformly distributed over the entire surface. The
tunneling current image should show a uniform
background, but this is not the case. The interaction
between tip and sample perturbs the electron density to
the extent that the tunneling current is slightly increased
when the tip is positioned directly above a surface atom.
The periodic array of atoms is clearly visible in the images
of materials such as gold, platinum, silver, nickel,
• Applications
• Several surfaces have been studied with the STM. The
arrangement of individual atoms on the metal surfaces
of gold, platinum, nickel, and copper have all been
accurately documented. The absorption
and diffusion of different species such as oxygen and
the epitaxial growth of gold on gold, silver on gold,
and nickel on gold also have been examined in detail.
• The surfaces of silicon have been studied more
extensively than those of any other material. The
surfaces are prepared by being heated in vacuum to
temperatures so high that the atoms there rearrange
their positions in a process called surface
reconstruction. The reconstruction of the silicon
surface designated (111) has been studied in minute
• “Vacuum tunneling” of electrons from tip to sample
can take place even though the environment in the
region surrounding the tip is not a vacuum but is filled
with molecules of gas or liquids. With a tip-sample
spacing as small as five angstroms, there is little room
for molecules—even though they may exist in the
surrounding atmosphere.
• The STM can operate in ambient atmosphere as well
as in high vacuum. Indeed, it has been operated in air,
in water, in insulating fluids, and in the ionic solutions
used in electrochemistry. It is much more convenient
than ultrahigh-vacuum instruments. When a high-
vacuum environment is employed, its purpose is not
to improve the performance of the STM but rather to
ensure the cleanliness of the sample surface.
• The STM can be cooled to temperatures less than 4 K (−269 °C,
or −452 °F)—the temperature of liquid helium. It can be heated
above 973 K (700 °C, or 1,300 °F). The low temperature is used
to investigate the properties of superconducting materials,
while the high temperature is employed to study the
rapid diffusion of atoms across the surface of metals and their
corrosion.
• The STM is used primarily for imaging, but there are many
other modalities that have been explored. The strong electric
field between tip and sample has been utilized to move atoms
along the sample surface. It has been used to enhance the
etching rates in various gases. In one instance, a voltage of four
volts was applied; the field at the tip was strong enough to
remove atoms from the tip and deposit them on a substrate.
This procedure has been employed with a gold tip to fabricate
small gold islands or clusters on the substrate with several
hundred atoms of gold in each cluster. These nanostructures are
used to pattern the surface on a scale that is unprecedented.
 Atomic force microscopy (AFM)
• Atomic force microscopy (AFM) or scanning force
microscopy (SFM) is a very-high-resolution type
of scanning probe microscopy (SPM), with
demonstrated resolution on the order of fractions of
a nanometer, more than 1000 times better than
the optical diffraction-limit.
• The information is gathered by "feeling" or "touching"
the surface with a mechanical
probe. Piezoelectric elements that facilitate tiny but
accurate and precise movements on (electronic)
command enable precise scanning.
• The atomic force microscope (AFM) was developed to
overcome a basic drawback with STM – it can only
image conducting or semiconducting surfaces. The
AFM has the advantage of imaging almost any type of
surface, including polymers, ceramics, composites,
glass, and biological samples.
• Binnig, Quate, and Gerber invented the AFM in 1985.
Their original AFM consisted of a diamond shard
attached to a strip of gold foil. The diamond tip
contacted the surface directly, with the interatomic van
der Waals forces providing the interaction mechanism.
Detection of the cantilever’s vertical movement was
done with a second tip – an STM placed above the
cantilever.
Typical configuration of
an AFM.
(1): Cantilever,
(2): Support for cantilever,
(3): Piezoelectric element (to
oscillate cantilever at its eigen
frequency),
(4): Tip (Fixed to open end of a
cantilever, acts as the probe),
(5): Detector of deflection and
motion of the cantilever
(6): Sample to be measured by
AFM,
(7): xyz drive, (moves sample (6) and
stage (8) in x, y, and z directions with
respect to a tip apex (4)), and
(8): Stage.
• The interaction between tip and sample, which can
be an atomic scale phenomenon, is transduced
into changes of the motion of cantilever which is a
macro scale phenomenon. Several different
aspects of the cantilever motion can be used to
quantify the interaction between the tip and
sample, most commonly the value of the
deflection, the amplitude of an imposed oscillation
of the cantilever, or the shift in resonance
frequency of the cantilever
• Detector
• The detector (5) of AFM measures the
deflection (displacement with respect to the
equilibrium position) of the cantilever and
converts it into an electrical signal. The intensity
of this signal will be proportional to the
displacement of the cantilever.
• Various methods of detection can be used, e.g.
interferometry, optical levers, the piezoelectric
method, and STM-based detectors (see section
"AFM cantilever deflection measurement")
 Working
• Analogous to how an Scanning Tunneling Microscope works, a
sharp tip is raster-scanned over a surface using a feedback
loop to adjust parameters needed to image a surface. Unlike
Scanning Tunneling Microscopes, the Atomic Force
Microscope does not need a conducting sample. Instead of
using the quantum mechanical effect of tunneling, atomic
forces are used to map the tip-sample interaction.
• Often referred to as scanning probe microscopy (SPM), there
are Atomic Force Microscopy techniques for almost any
measurable force interaction – van der Waals, electrical,
magnetic, thermal. For some of the more specialized
techniques, modified tips and software adjustments are
needed.
• In addition to Angstrom-level positioning and feedback loop
control, there are 2 components typically included in Atomic
Force Microscopy: Deflection and Force Measurement.
 AFM Probe Deflection
• Traditionally, most Atomic Force Microscopes use a
laser beam deflection system where a laser is
reflected from the back of the reflective AFM lever
and onto a position-sensitive detector. AFM tips and
cantilevers are typically micro-fabricated from Si or
Si3N4. Typical tip radius is from a few to 10s of nm.
• Measuring Forces
• Because the Atomic Force Microscope relies
on the forces between the tip and sample,
these forces impact AFM imaging. The force is
not measured directly, but calculated by
measuring the deflection of the lever, knowing
the stiffness of the cantilever.
• Hooke’s law gives:
F = -kz
• where F is the force, k is the stiffness of the
lever, and z is the distance the lever is bent.
 Feedback Loop for Atomic Force Microscopy
• Atomic Force Microscopy has a feedback loop using the
laser deflection to control the force and tip position. As
shown, a laser is reflected from the back of a cantilever
that includes the AFM tip. As the tip interacts with the
surface, the laser position on the photodetector is used
in the feedback loop to track the surface for imaging and
measuring.
• When using the AFM to image a sample, the tip is brought into
contact with the sample, and the sample is raster scanned along
an x–y grid .
• Most commonly, an electronic feedback loop is employed to keep
the probe-sample force constant during scanning. This feedback
loop has the cantilever deflection as input, and its output controls
the distance along the z axis between the probe support and the
sample support .
• As long as the tip remains in contact with the sample, and the
sample is scanned in the x–y plane, height variations in the
sample will change the deflection of the cantilever. The feedback
then adjusts the height of the probe support so that the
deflection is restored to a user-defined value (the setpoint).
• A properly adjusted feedback loop adjusts the support-sample
separation continuously during the scanning motion, such that the
deflection remains approximately constant. In this situation, the
feedback output equals the sample surface topography to within a
small error.