Quantum physics
"I think it is safe to say that no
one understands quantum mechanics."
Physicist Richard P. Feynman
To say that the invention of
semiconductor devices was a revolution would not be an exaggeration. Not
only was this an impressive technological accomplishment, but it paved
the way for developments that would indelibly alter modern society.
Semiconductor devices made possible miniaturized electronics, including
computers, certain types of medical diagnostic and treatment equipment,
and popular telecommunication devices, to name a few applications of
this technology.
But behind this revolution in technology
stands an even greater revolution in general science: the field of
quantum physics. Without this leap in understanding the natural
world, the development of semiconductor devices (and more advanced
electronic devices still under development) would never have been
possible. Quantum physics is an incredibly complicated realm of science,
and this chapter is by no means a complete discussion of it, but rather
a brief overview. When scientists of Feynman's caliber say that "no one
understands [it]," you can be sure it is a complex subject. Without a
basic understanding of quantum physics, or at least an understanding of
the scientific discoveries that led to its formulation, though, it is
impossible to understand how and why semiconductor electronic devices
function. Most introductory electronics textbooks I've read attempt to
explain semiconductors in terms of "classical" physics, resulting in
more confusion than comprehension.
Many of us have seen diagrams of atoms
that look something like this:
Tiny particles of matter called
protons and neutrons make up the center of the atom, while
electrons orbit around not unlike planets around a star. The nucleus
carries a positive electrical charge, owing to the presence of protons
(the neutrons have no electrical charge whatsoever), while the atom's
balancing negative charge resides in the orbiting electrons. The
negative electrons tend to be attracted to the positive protons just as
planets are gravitationally attracted toward whatever object(s) they
orbit, yet the orbits are stable due to the electrons' motion. We owe
this popular model of the atom to the work of Ernest Rutherford, who
around the year 1911 experimentally determined that atoms' positive
charges were concentrated in a tiny, dense core rather than being spread
evenly about the diameter as was proposed by an earlier researcher, J.J.
Thompson.
While Rutherford's atomic model accounted
for experimental data better than Thompson's, it still wasn't perfect.
Further attempts at defining atomic structure were undertaken, and these
efforts helped pave the way for the bizarre discoveries of quantum
physics. Today our understanding of the atom is quite a bit more
complex. However, despite the revolution of quantum physics and the
impact it had on our understanding of atomic structure, Rutherford's
solar-system picture of the atom embedded itself in the popular
conscience to such a degree that it persists in some areas of study even
when inappropriate.
Consider this short description of
electrons in an atom, taken from a popular electronics textbook:
Orbiting negative electrons are
therefore attracted toward the positive nucleus, which leads us to the
question of why the electrons do not fly into the atom's nucleus. The
answer is that the orbiting electrons remain in their stable orbit due
to two equal but opposite forces. The centrifugal outward force
exerted on the electrons due to the orbit counteracts the attractive
inward force (centripetal) trying to pull the electrons toward the
nucleus due to the unlike charges.
In keeping with the Rutherford model,
this author casts the electrons as solid chunks of matter engaged in
circular orbits, their inward attraction to the oppositely charged
nucleus balanced by their motion. The reference to "centrifugal force"
is technically incorrect (even for orbiting planets), but is easily
forgiven due to its popular acceptance: in reality, there is no such
thing as a force pushing any orbiting body away from its
center of orbit. It only seems that way because a body's inertia tends
to keep it traveling in a straight line, and since an orbit is a
constant deviation (acceleration) from straight-line travel, there is
constant inertial opposition to whatever force is attracting the body
toward the orbit center (centripetal), be it gravity, electrostatic
attraction, or even the tension of a mechanical link.
The real problem with this explanation,
however, is the idea of electrons traveling in circular orbits in the
first place. It is a verifiable fact that accelerating electric charges
emit electromagnetic radiation, and this fact was known even in
Rutherford's time. Since orbiting motion is a form of acceleration (the
orbiting object in constant acceleration away from normal, straight-line
motion), electrons in an orbiting state should be throwing off radiation
like mud from a spinning tire. Electrons accelerated around circular
paths in particle accelerators called synchrotrons are known to
do this, and the result is called synchrotron radiation. If
electrons were losing energy in this way, their orbits would eventually
decay, resulting in collisions with the positively charged nucleus.
However, this doesn't ordinarily happen within atoms. Indeed, electron
"orbits" are remarkably stable over a wide range of conditions.
Furthermore, experiments with "excited"
atoms demonstrated that electromagnetic energy emitted by an atom occurs
only at certain, definite frequencies. Atoms that are "excited" by
outside influences such as light are known to absorb that energy and
return it as electromagnetic waves of very specific frequencies, like a
tuning fork that rings at a fixed pitch no matter how it is struck. When
the light emitted by an excited atom is divided into its constituent
frequencies (colors) by a prism, distinct lines of color appear in the
spectrum, the pattern of spectral lines being unique to that element. So
regular is this phenomenon that it is commonly used to identify atomic
elements, and even measure the proportions of each element in a compound
or chemical mixture. According to Rutherford's solar-system atomic model
(regarding electrons as chunks of matter free to orbit at any radius)
and the laws of classical physics, excited atoms should be able to
return energy over a virtually limitless range of frequencies rather
than a select few. In other words, if Rutherford's model were correct,
there would be no "tuning fork" effect, and the light spectrum emitted
by any atom would appear as a continuous band of colors rather than as a
few distinct lines.
A pioneering researcher by the name of
Neils Bohr attempted to improve upon Rutherford's model after studying
in Rutherford's laboratory for several months in 1912. Trying to
harmonize the findings of other physicists (most notably, Max Planck and
Albert Einstein), Bohr suggested that each electron possessed a certain,
specific amount of energy, and that their orbits were likewise
quantized such that they could only occupy certain places around the
nucleus, somewhat like marbles fixed in circular tracks around the
nucleus rather than the free-ranging satellites they were formerly
imagined to be. In deference to the laws of electromagnetics and
accelerating charges, Bohr referred to these "orbits" as stationary
states so as to escape the implication that they were in motion.
While Bohr's ambitious attempt at
re-framing the structure of the atom in terms that agreed closer to
experimental results was a milestone in physics, it was by no means
complete. His mathematical analyses produced better predictions of
experimental events than analyses belonging to previous models, but
there were still some unanswered questions as to why electrons
would behave in such strange ways. The assertion that electrons existed
in stationary, quantized states around the nucleus certainly accounted
for experimental data better than Rutherford's model, but he had no idea
what would force electrons to manifest those particular states. The
answer to that question had to come from another physicist, Louis de
Broglie, about a decade later.
De Broglie proposed that electrons, like
photons (particles of light) manifested both particle-like and wave-like
properties. Building on this proposal, he suggested that an analysis of
orbiting electrons from a wave perspective rather than a particle
perspective might make more sense of their quantized nature. Indeed,
this was the case, and another breakthrough in understanding was
reached.
The atom according to de Broglie
consisted of electrons existing in the form of standing waves, a
phenomenon well known to physicists in a variety of forms. Like the
plucked string of a musical instrument vibrating at a resonant
frequency, with "nodes" and "antinodes" at stable positions along its
length, de Broglie envisioned electrons around atoms standing as waves
bent around a circle:
Electrons could only exist in certain,
definite "orbits" around the nucleus because those were the only
distances where the wave ends would match. In any other radius, the wave
would destructively interfere with itself and thus cease to exist.
De Broglie's hypothesis gave both
mathematical support and a convenient physical analogy to account for
the quantized states of electrons within an atom, but his atomic model
was still incomplete. Within a few years, though, physicists Werner
Heisenberg and Erwin Schrodinger, working independently of each other,
built upon de Broglie's concept of a matter-wave duality to create more
mathematically rigorous models of subatomic particles.
This theoretical advance from de
Broglie's primitive standing wave model to Heisenberg's matrix and
Schrodinger's differential equation models was given the name quantum
mechanics, and it introduced a rather shocking characteristic to the
world of subatomic particles: the trait of probability, or uncertainty.
According to the new quantum theory, it was impossible to determine the
exact position and exact momentum of a particle at the same time.
Popular explanations of this "uncertainty principle" usually cast it in
terms of error caused by the process of measurement (i.e. by attempting
to precisely measure the position of an electron, you interfere with its
momentum and thus cannot know what it was before the position
measurement was taken, and visa versa), but the truth is actually much
more mysterious than simple measurement interference. The startling
implication of quantum mechanics is that particles do not actually
possess precise positions and momenta, but rather balance the two
quantities in a such way that their combined uncertainties never
diminish below a certain minimum value.
It is interesting to note that this form
of "uncertainty" relationship exists in areas other than quantum
mechanics. As discussed in the "Mixed-Frequency AC Signals" chapter in
volume II of this book series, there is a mutually exclusive
relationship between the certainty of a waveform's time-domain data and
its frequency-domain data. In simple terms, the more precisely we know
its constituent frequency(ies), the less precisely we know its amplitude
in time, and visa-versa. To quote myself:
A waveform of infinite duration
(infinite number of cycles) can be analyzed with absolute precision,
but the less cycles available to the computer for analysis, the less
precise the analysis. . . The fewer times that a wave cycles, the less
certain its frequency is. Taking this concept to its logical extreme,
a short pulse -- a waveform that doesn't even complete a cycle --
actually has no frequency, but rather acts as an infinite range of
frequencies. This principle is common to all wave-based phenomena, not
just AC voltages and currents.
In order to precisely determine the
amplitude of a varying signal, we must sample it over a very narrow span
of time. However, doing this limits our view of the wave's frequency.
Conversely, to determine a wave's frequency with great precision, we
must sample it over many, many cycles, which means we lose view of its
amplitude at any given moment. Thus, we cannot simultaneously know the
instantaneous amplitude and the overall frequency of any wave with
unlimited precision. Stranger yet, this uncertainty is much more than
observer imprecision; it resides in the very nature of the wave itself.
It is not as though it would be possible, given the proper technology,
to obtain precise measurements of both instantaneous amplitude
and frequency at once. Quite literally, a wave cannot possess both a
precise, instantaneous amplitude, and a precise frequency at the same
time.
Likewise, the minimum uncertainty of a
particle's position and momentum expressed by Heisenberg and Schrodinger
has nothing to do with limitation in measurement; rather it is an
intrinsic property of the particle's matter-wave dual nature. Electrons,
therefore, do not really exist in their "orbits" as precisely defined
bits of matter, or even as precisely defined waveshapes, but rather as
"clouds" -- the technical term is wavefunction -- of probability
distribution, as if each electron were "spread" or "smeared" over a
range of positions and momenta.
This radical view of electrons as
imprecise clouds at first seems to contradict the original principle of
quantized electron states: that electrons exist in discrete, defined
"orbits" around atomic nuclei. It was, after all, this discovery that
led to the formation of quantum theory to explain it. How odd it seems
that a theory developed to explain the discrete behavior of electrons
ends up declaring that electrons exist as "clouds" rather than as
discrete pieces of matter. However, the quantized behavior of electrons
does not depend on electrons having definite position and momentum
values, but rather on other properties called quantum numbers. In
essence, quantum mechanics dispenses with commonly held notions of
absolute position and absolute momentum, and replaces them with absolute
notions of a sort having no analogue in common experience.
Even though electrons are known to exist
in ethereal, "cloud-like" forms of distributed probability rather than
as discrete chunks of matter, those "clouds" possess other
characteristics that are discrete. Any electron in an atom can be
described in terms of four numerical measures (the previously mentioned
quantum numbers), called the Principal, Angular
Momentum, Magnetic, and Spin numbers. The following is
a synopsis of each of these numbers' meanings:
Principal Quantum Number:
Symbolized by the letter n, this number describes the shell
that an electron resides in. An electron "shell" is a region of space
around an atom's nucleus that electrons are allowed to exist in,
corresponding to the stable "standing wave" patterns of de Broglie and
Bohr. Electrons may "leap" from shell to shell, but cannot exist
between the shell regions.
The principle quantum number can be any
positive integer (a whole number, greater than or equal to 1). In other
words, there is no such thing as a principle quantum number for an
electron of 1/2 or -3. These integer values were not arrived at
arbitrarily, but rather through experimental evidence of light spectra:
the differing frequencies (colors) of light emitted by excited hydrogen
atoms follow a sequence mathematically dependent on specific, integer
values.
Each shell has the capacity to hold
multiple electrons. An analogy for electron shells is the concentric
rows of seats of an amphitheater. Just as a person seated in an
amphitheater must choose a row to sit in (for there is no place to sit
in the space between rows), electrons must "choose" a particular
shell to "sit" in. Like amphitheater rows, the outermost shells are able
to hold more electrons than the inner shells. Also, electrons tend to
seek the lowest available shell, like people in an amphitheater trying
to find the closest seat to the center stage. The higher the shell
number, the greater the energy of the electrons in it.
The maximum number of electrons that any
shell can hold is described by the equation 2n2, where "n" is
the principle quantum number. Thus, the first shell (n=1) can hold 2
electrons; the second shell (n=2) 8 electrons, and the third shell (n=3)
18 electrons.
Electron shells in an atom are sometimes
designated by letter rather than by number. The first shell (n=1) is
labeled K, the second shell (n=2) L, the third shell (n=3) M, the fourth
shell (n=4) N, the fifth shell (n=5) O, the sixth shell (n=6) P, and the
seventh shell (n=7) Q.
Angular Momentum Quantum Number:
Within each shell, there are subshells. One might be inclined to
think of subshells as simple subdivisions of shells, like lanes dividing
a road, but the truth is much stranger than this. Subshells are regions
of space where electron "clouds" are allowed to exist, and different
subshells actually have different shapes. The first subshell is
shaped like a sphere, which makes sense to most people, visualizing a
cloud of electrons surrounding the atomic nucleus in three dimensions.
The second subshell, however, resembles a dumbbell, comprised of two
"lobes" joined together at a single point near the atom's center. The
third subshell typically resembles a set of four "lobes" clustered
around the atom's nucleus. These subshell shapes are reminiscent of
graphical depictions of radio antenna signal strength, with bulbous
lobe-shaped regions extending from the antenna in various directions.
Valid angular momentum quantum numbers
are positive integers like principal quantum numbers, but also include
zero. These quantum numbers for electrons are symbolized by the letter
l. The number of subshells in a shell is equal to the shell's
principal quantum number. Thus, the first shell (n=1) has one subshell,
numbered 0; the second shell (n=2) has two subshells, numbered 0 and 1;
the third shell (n=3) has three subshells, numbered 0, 1, and 2.
An older convention for subshell
description used letters rather than numbers. In this notational system,
the first subshell (l=0) was designated s, the second subshell
(l=1) designated p, the third subshell (l=2) designated d,
and the fourth subshell (l=3) designated f. The letters come from
the words sharp, principal (not to be confused with the
principal quantum number, n), diffuse, and fundamental.
You will still see this notational convention in many periodic tables,
used to designate the electron configuration of the atoms' outermost, or
valence, shells.
Magnetic Quantum Number: The
magnetic quantum number for an electron classifies which orientation its
subshell shape is pointed. For each subshell in each shell, there are
multiple directions in which the "lobes" can point, and these different
orientations are called orbitals. For the first subshell (s;
l=0), which resembles a sphere, there is no "direction" it can "point,"
so there is only one orbital. For the second (p; l=1) subshell in each
shell, which resembles a dumbbell, there are three different directions
they can be oriented (think of three dumbbells intersecting in the
middle, each oriented along a different axis in a three-axis coordinate
system).
Valid numerical values for this quantum
number consist of integers ranging from -l to l, and are symbolized as
ml in atomic physics and lz in
nuclear physics. To calculate the number of orbitals in any given
subshell, double the subshell number and add 1 (2l + 1). For example,
the first subshell (l=0) in any shell contains a single orbital,
numbered 0; the second subshell (l=1) in any shell contains three
orbitals, numbered -1, 0, and 1; the third subshell (l=2) contains five
orbitals, numbered -2, -1, 0, 1, and 2; and so on.
Like principal quantum numbers, the
magnetic quantum number arose directly from experimental evidence: the
division of spectral lines as a result of exposing an ionized gas to a
magnetic field, hence the name "magnetic" quantum number.
Spin Quantum Number: Like the
magnetic quantum number, this property of atomic electrons was
discovered through experimentation. Close observation of spectral lines
revealed that each line was actually a pair of very closely-spaced
lines, and this so-called fine structure was hypothesized to be
the result of each electron "spinning" on an axis like a planet.
Electrons with different "spins" would give off slightly different
frequencies of light when excited, and so the quantum number of "spin"
came to be named as such. The concept of a spinning electron is now
obsolete, being better suited to the (incorrect) view of electrons as
discrete chunks of matter rather than as the "clouds" they really are,
but the name remains.
Spin quantum numbers are symbolized as
ms in atomic physics and sz in nuclear
physics. For each orbital in each subshell in each shell, there can be
two electrons, one with a spin of +1/2 and the other with a spin of
-1/2.
The physicist Wolfgang Pauli developed a
principle explaining the ordering of electrons in an atom according to
these quantum numbers. His principle, called the Pauli exclusion
principle, states that no two electrons in the same atom may occupy
the exact same quantum states. That is, each electron in an atom has a
unique set of quantum numbers. This limits the number of electrons that
may occupy any given orbital, subshell, and shell.
Shown here is the electron arrangement
for a hydrogen atom:
With one proton in the nucleus, it takes
one electron to electrostatically balance the atom (the proton's
positive electric charge exactly balanced by the electron's negative
electric charge). This one electron resides in the lowest shell (n=1),
the first subshell (l=0), in the only orbital (spatial orientation) of
that subshell (ml=0), with a spin value of 1/2. A very common
method of describing this organization is by listing the electrons
according to their shells and subshells in a convention called
spectroscopic notation. In this notation, the shell number is shown
as an integer, the subshell as a letter (s,p,d,f), and the total number
of electrons in the subshell (all orbitals, all spins) as a superscript.
Thus, hydrogen, with its lone electron residing in the base level, would
be described as 1s1.
Proceeding to the next atom type (in
order of atomic number), we have the element helium:
A helium atom has two protons in the
nucleus, and this necessitates two electrons to balance the
double-positive electric charge. Since two electrons -- one with
spin=1/2 and the other with spin=-1/2 -- will fit into one orbital, the
electron configuration of helium requires no additional subshells or
shells to hold the second electron.
However, an atom requiring three or more
electrons will require additional subshells to hold all
electrons, since only two electrons will fit into the lowest shell
(n=1). Consider the next atom in the sequence of increasing atomic
numbers, lithium:
An atom of lithium only uses a fraction
of the L shell's (n=2) capacity. This shell actually has a total
capacity of eight electrons (maximum shell capacity = 2n2
electrons). If we examine the organization of the atom with a completely
filled L shell, we will see how all combinations of subshells, orbitals,
and spins are occupied by electrons:
Often, when the spectroscopic notation is
given for an atom, any shells that are completely filled are omitted,
and only the unfilled, or the highest-level filled shell, is denoted.
For example, the element neon (shown in the previous illustration),
which has two completely filled shells, may be spectroscopically
described simply as 2p6 rather than 1s22s22p6.
Lithium, with its K shell completely filled and a solitary electron in
the L shell, may be described simply as 2s1 rather than 1s22s1.
The omission of completely filled,
lower-level shells is not just a notational convenience. It also
illustrates a basic principle of chemistry: that the chemical behavior
of an element is primarily determined by its unfilled shells. Both
hydrogen and lithium have a single electron in their outermost shells
(1s1 and 2s1, respectively), and this gives the
two elements some similar properties. Both are highly reactive, and
reactive in much the same way (bonding to similar elements in similar
modes). It matters little that lithium has a completely filled K shell
underneath its almost-vacant L shell: the unfilled L shell is the shell
that determines its chemical behavior.
Elements having completely filled outer
shells are classified as noble, and are distinguished by their
almost complete non-reactivity with other elements. These elements used
to be classified as inert, when it was thought that they were
completely unreactive, but it is now known that they may form compounds
with other elements under certain conditions.
Given the fact that elements with
identical electron configurations in their outermost shell(s) exhibit
similar chemical properties, it makes sense to organize the different
elements in a table accordingly. Such a table is known as a periodic
table of the elements, and modern tables follow this general form:
Dmitri Mendeleev, a Russian chemist, was
the first to develop a periodic table of the elements. Although
Mendeleev organized his table according to atomic mass rather than
atomic number, and so produced a table that was not quite as useful as
modern periodic tables, his development stands as an excellent example
of scientific proof. Seeing the patterns of periodicity (similar
chemical properties according to atomic mass), Mendeleev hypothesized
that all elements would fit into this ordered scheme. When he discovered
"empty" spots in the table, he followed the logic of the existing order
and hypothesized the existence of heretofore undiscovered elements. The
subsequent discovery of those elements granted scientific legitimacy to
Mendeleev's hypothesis, further discoveries leading to the form of the
periodic table we use today.
This is how science should work:
hypotheses followed to their logical conclusions, and accepted,
modified, or rejected as determined by the agreement of experimental
data to those conclusions. Any fool can formulate a hypothesis
after-the-fact to explain existing experimental data, and many do. What
sets a scientific hypothesis apart from post hoc speculation is
the prediction of future experimental data yet uncollected, and the
possibility of disproof as a result of that data. To boldly follow a
hypothesis to its logical conclusion(s) and dare to predict the results
of future experiments is not a dogmatic leap of faith, but rather a
public test of that hypothesis, open to challenge from anyone able to
produce contradictory data. In other words, scientific hypotheses are
always "risky" in the sense that they claim to predict the results of
experiments not yet conducted, and are therefore susceptible to disproof
if the experiments do not turn out as predicted. Thus, if a hypothesis
successfully predicts the results of repeated experiments, there is
little probability of its falsehood.
Quantum mechanics, first as a hypothesis
and later as a theory, has proven to be extremely successful in
predicting experimental results, hence the high degree of scientific
confidence placed in it. Many scientists have reason to believe that it
is an incomplete theory, though, as its predictions hold true more so at
very small physical scales than at macroscopic dimensions, but
nevertheless it is a tremendously useful theory in explaining and
predicting the interactions of particles and atoms.
As you have already seen in this chapter,
quantum physics is essential in describing and predicting many different
phenomena. In the next section, we will see its significance in the
electrical conductivity of solid substances, including semiconductors.
Simply put, nothing in chemistry or solid-state physics makes sense
within the popular theoretical framework of electrons existing as
discrete chunks of matter, whirling around atomic nuclei like miniature
satellites. It is only when electrons are viewed as "wavefunctions"
existing in definite, discrete states that the regular and periodic
behavior of matter can be explained.
- REVIEW:
- Electrons in atoms exist in "clouds"
of distributed probability, not as discrete chunks of matter orbiting
the nucleus like tiny satellites, as common illustrations of atoms
show.
- Individual electrons around an atomic
nucleus seek unique "states," described by four quantum numbers:
the Principal Quantum Number, otherwise known as the shell;
the Angular Momentum Quantum Number, otherwise known as the
subshell; the Magnetic Quantum Number, describing the
orbital (subshell orientation); and the Spin Quantum Number,
or simply spin. These states are quantized, meaning that there
are no "in-between" conditions for an electron other than those states
that fit into the quantum numbering scheme.
- The Principal Quantum Number (n)
describes the basic level or shell that an electron resides in. The
larger this number, the greater radius the electron cloud has from the
atom's nucleus, and the greater than electron's energy. Principal
quantum numbers are whole numbers (positive integers).
- The Angular Momentum Quantum Number
(l) describes the shape of the electron cloud within a
particular shell or level, and is often known as the "subshell." There
are as many subshells (electron cloud shapes) in any given shell as
that shell's principal quantum number. Angular momentum quantum
numbers are positive integers beginning at zero and terminating at one
less than the principal quantum number (n-1).
- The Magnetic Quantum Number (ml)
describes which orientation a subshell (electron cloud shape) has.
There are as many different orientations for each subshell as the
subshell number (l) plus 1, and each unique orientation is
called an orbital. These numbers are integers ranging from the
negative value of the subshell number (l) through 0 to the
positive value of the subshell number.
- The Spin Quantum Number (ms)
describes another property of an electron, and can be a value of +1/2
or -1/2.
- Pauli's Exclusion Principle
says that no two electrons in an atom may share the exact same set of
quantum numbers. Therefore, there is room for two electrons in each
orbital (spin=1/2 and spin=-1/2), 2l+1 orbitals in every
subshell, and n subshells in every shell, and no more.
- Spectroscopic notation is a
convention for denoting the electron configuration of an atom. Shells
are shown as whole numbers, followed by subshell letters (s,p,d,f),
with superscripted numbers totaling the number of electrons residing
in each respective subshell.
- An atom's chemical behavior is solely
determined by the electrons in the unfilled shells. Low-level shells
that are completely filled have little or no effect on the chemical
bonding characteristics of elements.
- Elements with completely filled
electron shells are almost entirely unreactive, and are called
noble (formerly known as inert).
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