- Dec 2023
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Local file Local file
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homotopy equivalent to Sn i
What does homotopy equivalent mean?
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geocalc.clas.asu.edu geocalc.clas.asu.edu
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International Conference on Clif-ford Algebras and their Applications in Mathematical Physics”
Oooh. Is this still ongoing?
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discussed in the sequel to this paper.
Check that out.
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Local file Local file()21
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metric product
I thought metrics don't define a scalar product, only that the reverse is true.
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Bivectorschange their sign
makes sense as reversing the order of the wedge product. What does it mean in the context of reversing order of geometric product?
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∧rB
How is this defined? Because I thought that this symbol is defined only for vector spaces.?
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2 = (−1) d(d−1)2 +s,
Where does this come from.
-
I = e1 . . . ed = e1 ∧ . . . ∧ e
Is the product without wedges just notation, or does it mean geometric product?
-
IR
I presume it's really just whatever field the vector space V is over.
-
polyvector
What is a poly-vector? it seems undefined.
-
Clifford algebra structure may be defined in an abstract way, with aClifford product. In this case, the vector space of multivectors is simply a peculiar repre-sentation.
seems to be what AV8N promulgates.
-
(bilateral) ideal
What is an "ideal"?
-
(In the special case where themetric is zero, Cℓ(V ) = ∧ V .)
Why does this hold?
-
One defines the Clifford (or geometrical) product of two vectors asu v ≡ u · v + u ∧ v.
only works for vectors.
-
The wedge product of a p−mutivector by itself, M ∧ M , is always 0 when p isodd. This is not true when p is even
What happens when p is even? Given that the Wedge product is completely antisymetric, M^M = - M^M And this can only be true if M^M =0
Why do they say not true when P is even?
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form ǫαβγ.
is this really a multi-form?
-
canonical (musical) isomorphism between V and its dual V ∗, which extends to theexterior algebras ∧ V and ∧ V ∗;
any basis for V defines a basis for v*, right? combined with this statement, wouldn't that mean that inner product defines a basis? or something similar?
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an (Hodge) duality (1.2.4) in the exterior algebras
I thought this was just like picking a basis.
-
exterior algebra [of multi-vectors] ∧ V of V
This implies there are other exterior algebras.
-
⊗
is this what the geometric product is?
-
e defines the vector space of all tensors
in this case you can have a vector plus a scalar ... rather odd. I thought that was unique to clifford algebra, rather than being an aspect of tensors.
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normalized)
the 1/2 is what they mean by normalization?
-
τ (s,r)V ≡s⊗ V ∗r⊗ V
are the vector space V and V* also tensored together? Because there is a tensor product over r and one over s, but they are just listed next to eachother.
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An antisymmetric [con-travariant]tensor of type (0; p) will be called a p-vector , more generally a multivector .An antisymmetric [covariant] tensor of type (p; 0) defines a p-form, more generally amultiform (more simply, a form
So, anti-symmetry = vector? Why?
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- Nov 2023
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journals.aps.org journals.aps.org
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that the separation of the spin componentsby the magnetic interaction is counteracted by the effect ofthe Lorentz force on the moving particle.
How can that be true, because 1. Stern Gerloch 2. If the electron had opposite sign, the counteraction would be working together. By symmetry, changing the electron charge should not result in different physics of whether you can tell apart electron spins or not.
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Themagnetic moment of the orbital motion of the electron in anatom is of the same order, but it can be taken to a classicalvalue by increasing the angular momentum quantum num-ber.
How is it increased? And what do they mean by it is of the same order if it can be increased?
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Local file Local file
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the situation is different for X-ray MO effects. Here the magnetic circular dichroism, whichis equivalent to the Faraday ellipticity, is a commonly used technique
why is mcd more popular for xray? seems also that this is done on reflection, because the author compares it to Faraday ellipticity, which is on transmission.
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Both z { and } { are thereby expressed as an angle
which is approximately b/a as you were using.
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Various experimental techniques for detecting the Kerr rotation and ellipticity have beendiscussed in the literature (see, e.g., Robinson 1963, Jasperson and Schnatterly 1969, Suits1971, Sato 1981). Schoenes (1992) has given a classification of the available techniquesand a discussion of their respective advantages and disadvantages.
Worth reading
-
m first-principles energy-bandtheory, and even beyond, that it is feasible to make ab initio predictions of MOKE spectra
What's the difference between this ab initio, versus the "first principles"?
-
effect of Fe provides a minute description of theexperimental MOKE
minute as in computationally easy? or minute as in not fully accurate?
-
UNiSn is an antiferromagnetic material
Were the predicted kerr rotation for the antifero phase, or for the ferromagnetic phase.
-
m. Daalderop etal. (1988a) predicted a large Kerr rotation of 5j for UNiSn, but did, unfortunately, notpublish a test of the computational method on simpler systems like the elemental 3i metals.
Did they make their code available for others to do the test?
-
Another goal of MO research whichbecame intensively pursued in the eighties was to extracted information on the electronicstructure of, in particular, lanthanide and actinide compounds.
Why?
-
Most of the accumulatedexperimental MO data have recently been surveyed by Buschow (1988) and by Schoenes(1992),
Worth looking at.
-
measured MO spectraof many materials appeared in the last three decades.
So,,, frequency sweeps. ? Or what spectra?
-
become customary to relateMO phenomena to the materials band structure.
What Kelly was saying.
-
microwave magneto-absorption and Faradayeffect in semiconductors (Dresselhaus et al. 1955, Zwerdling and Lax 1957, Lax et al.1959).
Microwave moke is a thing!!!
-
nterest in MO recording, which since then has developed into a leading technologicalapplication of MOKE (
not recording, should be read out right?
-
to visualizing surface and subsurface magnetic domains developed.
How to see subsurface magnetic domains?
-
The only exception was the discovery of the MO Kerr effect in paramagneticmetals in an applied field by Majorana (1944
so the paramagnetic materials happened later ...
-
by Lorentz (1884), based on the idea that left- and right-circularly polarizedlight coupled differently t
Might be interesting to see this classical model.
-
MO phenomena hat become an important research topic.Quantum mechanics had not yet emerged, therefore the theoretical understanding of thephenomena was completely lacking
Implies that Magneto Optics is inherently quantum.
-
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arxiv.org arxiv.org
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time reversal is preservedand inversion symmetry is absent, a nonequilibrium Kerreffect is allowed by symmetry,
Why is that allowed by symmetry? What symmetry?
-
Without loss of generality,
How is polar WLOG?
-
both effects only require inversionsymmetry breaking
why inversion symmetry breaking?
-
nonequilibrium
Why nonequilibrium? Dynamic equilibrium when constant current?
Is there static equilibrium orbital magnets?
-
d that the rectificationinduced by the nonlinear Hall effect
How does the nonlinear Hall effect rectify anything?
-
Berrycurvature dipole [5] (BCD)
What is that?
-
a nonlinearHall effect
Does it have to be nonlinear, to get this effect?
-
even in the absence of spin-orbitcoupling
How do we know that?
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Local file Local file
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nondestructive techniques
non destructive in what way?
-
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Local file Local file
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his showsthat h is 2π-periodic, and therefore of the form h(t) g(c(t))
How does that follow?
-
R if k 0,0 if k ≥ 1.Proof. This is a restatement of the Poincaré lemma,
The real numbers for k=1 doesn't appear to stem from the point-care lemma.
-
ne can check that these operations arewell-defined
What does it mean to check well defined?
-
A contractible manifold is simply connected.
is the converse true?
-
k , 1
Isn't any 1-form also exact?, because you can slip it around the origin in a higher dimensional space?
-
µ µM be the volume form of M.
So, the existance of a volume form means that the manifold is not contractable. How doesn't that make everything non contractible?
-
∗dx nµM .
How does hodge dx know which way the unit normal is pointing?
-
α0 x · ∗dx‖x‖n .
Why to the nth power?
-
t dxi ),
Why did this term drop out in the next line? Something about the definition of kappa, I presume.
-
φ∗1 (α) α and φ∗0 (α) 0,
Not immediately obvious to me.
-
contraction ofM onto x0
Is this definition equivalent to being contractible at any point?
-
On somemanifolds all closed forms (of positive degree) are exact, on others this is true onlyin certain degrees
of zero degree (or negative) would also be exact no?
-
∫M φ∗ (α0 ) is 2π
could this be written as integral over mbar, where mbar is the embedded version of M in N?
-
0 ∫M×[0,1]φ∗ (dα)
Why is that zero?
-
ι0 (x)
What is iota in this case?
-
g dt dxJ , then ι∗0 (α) ι∗1 (α) 0
Intuitively zero, but rigorously not sure.
-
( f (x, 1) − f (x, 0)) dxI ι∗1 (α) − ι∗0 (α)
Why are these the pulbacks of alpha?
-
It can be regarded as an application of Stokes’ theorem, but weshall give a direct proof.
How to prove this from stokes?
-
by taking the piece of α involving dt and integrating it over the unit interval
Why is it defined that way?
-
exists a pointx0 in M
What is the manifold N in this case? Does the definition of contractible require N = M?
-
homotopy of loops from φ0 to φ1 in the punctured plane
Where the punctured plane is both M and N? What about maps from the punctured plane to the non punctured plane.
-
a piece of string moving through the manifold N
These strings could in general intersect, right?
-
A manifold M is said to be contractible
What is the intuition for contractible?
-
Suppose that φ0 and φ1 are two maps from amanifold M to a manifold N
If it requires an atlas, how do we know that we can get from N to M with one map?
-
φ(x, t) (1 − t)x + tx/‖x‖.
can you not do something like this for any two functions phi? Then everything would be homotopic.
-
φ0 (x) x (identity map) and φ1 (x) 0
feel very different in character. one is singular (non invertible) and the other is not.
Homotopy vs Homeomorphic?
-
he theorem also remains valid if the closed ball is replaced by aclosed cube or a similar shape.
Will we derive this with Homotopy?
-
∂M with the induced orientation
Was the point of the orientable manifold and the induced orientation just that it allows us to use stokes theorem?
-
so dβ 0.
is that true on M though?
-
φ∗ (β) be itspullback to M
What does it mean to have a volume form acting on a space which is one dimension higher?
-
(relative to some embedding of M into RN )
Can you have a volume form not referenced to an embedding?
-
orientable manifol
Are we assuming that the map phi must preserve orientation?
-
subset A
subset of the manifold or a subset of R^n
-
F(x) · n(x) (−1)N+1FN (x)
Definition of F_N?
-
F · ∗dx (F · n)µM .
Wouldn't the left hand side be a sum of n-1 forms? The right however seems to be volume forms. Whta gives?
-
n this situation it is best to think of F as the flow vectorfield of a fluid, where the direction of F(x) gives the direction of the flow
I believe it, but that's because I know the answer from physics already. Why is this the case?
-
α F · dx
is this really a dot product?
-
a subordinate partition ofunity consisting of functions
Why is this any easier than just splitting up the domain?
-
dti gi (t1, . . . , bi , . . . , tn ) − gi (t1 , . . . , ai , . . . , tn ) 0
how does this follow from the partial derivatives vanishing along the walls? Because this integral is going through the interior of the box.
-
Suppose M Hn . Then we can writeα n∑i1gi dt1 dt2 · · ·̂ dti · · · d
alpha seems to be an n-1 form, which is wrong because we'd have to integrate it against an n dimensional manifold.
-
64 5. INTEGRATION AND STOKES’ THEOREM5.1. Theorem. Let α be a k-form on U and c : R → U a smooth map. Let p : ¯R → Rbe a reparametrization. Then∫c◦pα ∫c α if p preserves the orientation,− ∫c α if p reverses the orientation
why no determinant?
-
e support of α is certainly compact if the manifold M itself is compact.
How do we know that that's true?
-
contained in ψi (Ui )
why contained within, and not identical to?
-
i) λi ≥ 0 for all
for all x in m? what does it mean to say that the function is positive?
-
breaking thedifferential form into small pieces a
what does that mean?
-
φm (x) ≥ cm .
So it's only 1 inequality? Can it be more?
-
em
What is em? is it an arbitrary m dimensional vector?
-
It follows that on thehypersurface M the Rn -valued n − 1-form ∗dx is equal to the product of n and ascalar n − 1-form ν: ∗dx nν.
How does that follow?
-
µM (e1, . . . , eN−1)
seems tp rely on the normalization assumption?
-
) (−1)N+1FN .
i is summed over. How does that follow?
-
form satisfies µM,x (e1 , . . . , eN−1) 1.
Assumes the vectors are normalized?
-
f (v1 , v2 , . . . , vn ; 1) is a negatively oriented frame
Defined based on the standard basis?
-
volume form µM depends on the embedding of M into RN . I
How are we to think about this in the coordinate independent way?
-
If this frame is positively,resp. negatively, oriented
Positively or negatively oriented with respect to what? The standard basis? This question comes into play because you cannot compare to the original frame if V and W are different vector spaces.
-
This defines an n-form ω on M and we must show that ω µM
What's different about this definition? It seems like the burden is not to show that they are equal, but rather that it is indeed an n-form?
-
he support of α is defined as the set of all points x in M with theproperty that for every open ball B around x there is a y ∈ B ∩ M such that αy , 0
Why do we have to play such tricks?
-
We define an orientationon M by requiring ψ to be orientation-preserving.
How do you require this after the fact?
-
orientation induced
how is this induced?
-
µM depends on the embedding of M into RN . It changes if we dilateor shrink or otherwise deform M
is this surprising?
-
volume forms are seldom exact!
why not exact? And why do the old fashioned notations make them seem exact?
-
det(Dφ(u) du1 du2 · · · dun
why does the determinant factor through?
-
µi ψ∗i (µ) √det(Dψi (t)T Dψi (t)) dt1 dt2 · · · dtn
Why?
-
Gauss map of M
not one to one.
-
N+1
should technically be N-1, but it's equivalent.
-
[n, e1, e2, . . . , eN−1; 1) [e1, e2, . . . , eN ; 1].
weird to talk about orientations of different dimensions. The change of basis matrix A is no longer invertible, no longer square, and no longer has a determinant.
How does this work out?
-
Proof
How do we know that this unit normal field defines a well defined frame over the entire manifold?
-
Choose ε ±1 such that (n(x), b1, b2, . . . , bn ; ε) is a positively orientedframe of RN .
positively oriented with respect to the standard basis?
-
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www.weylmann.com www.weylmann.com
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structure constant of the Pauli algebra,
What does that mean?
-
ut for mathematical reasons 2 × 2 matrices inphysics need to be unitary
not obvious to me why.
-
-
en.wikipedia.org en.wikipedia.org
-
The space of spinors is formally defined as the fundamental representation of the Clifford algebra. (
Wild.
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Vπ = 1.3 volts (EOSPACE Inc,Redmond, WA)
Must be a fiber EOM
-
The transmissionaxis (TA) of PL is aligned to the initial linear polarization of the optical beam and to the slow-axis
Odd. Why would they do that?
-
-
journals.aps.org journals.aps.orguntitled7
-
quation (6) is the fundamental result of this Letter. It
never used the fact that light must be decomposed to higher order in molecule size. They used just P and M in terms of polarizability.
Maybe these response tensors implicitly assume a finite size.
-
! Imð ~E ~BÞ ¼ _B E _E B
where did the w go?
-
E is odd under parity whileB is even
Why?
-
We consider a pairof such fields which are interchanged by application ofparity:~EðtÞ ¼ ~E0ei!t
I assume the pair here is the +-
Why isn't B interchanged by parity?
-
Yet chiral interac-tions require a time-even pseudoscalar,
Why is that?
-
does not occur within the point electricdipole approximation, but requires expansion to first orderin ka 103, where k is the wave vector of the light and ais the size of the molecule. I
Dont rotating dipoles output circularly polarized light?
Wouldn't that produce dcircular dicrhoism?
-
introduce a measure of the local density of chirality of the electromagnetic field.
Is that not just the handedness of light?
-
-
www.science.org www.science.org
-
ow w0¶ in which negative refractioncan be seen for one of the polarizations.
w = k/c That's positive everywhere here it seems. Only the group velocity is negative near w0' and w0'' ?
-
pened up where the permittivity is negative
Isn't the permittivity one value? What do they mean where it is negative?
-
resonant electric dipoles
resonant to what
-
Also, the bands at this point havefinite group velocity but infinite phasevelocity
Trying to figure out how that works.
-
Although referred to as Bleft-handed[ mate-rials, I stress that the sense in which thisterm was used has nothing to do with chi-rality. Therefore I prefer to use the expressionBnegatively refracting[ to avoid confusion.
Why do they call it that?
-
In addition, there are twolongitudinal modes (not shown), one magneticin character and the other electric.
That goes against standard wisdom!
-
-
web.mit.edu web.mit.edu
-
analogy is less perfect,
What analogy is less perfect?
-
magneticplasma
What is a magnetic plasma?
-
The underlying secret of this medium is that both the di-electric function, ´, and the magnetic permeability, m, hap-pen to be negative.
Is that required for both to be negative to get a negative refractive index?
-
-
-
lthough well-established textbook arguments suggest that static electricsusceptibilityχ(0) must be positive in “all bodies,
Which textbook arguments?
-
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Local file Local file
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The schematic layout (top) and false-coloured microscopeimage (bottom) of a typical device for photocapacitance measurements. Scalebar, 200 µm.
The gold doesn't seem to overlap the other piece of gold in the microscope image, whereas the schematic it does.
-
ed as a function of the X-ray power density forthe CsPbBr3 (002) peak.
Doesn't seem to line up with the plot b. There it seems like theta is on the order of 0.1 degree. However the FWHM is 10^-20
-
gle with an increasing X-ray powerdensity and indicating more local lattice distortion with increasing chargecarrier generation.
So, x ray induced photoionization. Why not do the laser photo ionization?
-
steps at 20 min intervals, and the illumination power values in the plot haveunits of mW cm −2
Seems like higher power doesn't have much of an effect. Question is if low power illumination for longer achieves same effect as the high illumination for shorter.
-
18.8 mW cm −2
Need to calculate to compare with what we have.
-
The dashed lines show fits to the photoconductance decay ateach temperature.
Seems to fit a bit too well, given the non exponential appearence.
-
at 10 K and 200 K is 1.0 × 10 6 s and 7.8 × 10 2 s
That assumes exponential decay. The plot doesn't look particularly exponential.
-
hows no apparent sign of decay during themeasurement timescale
What temperature is it at? can you change the temperature to change make the carriers die faster? Also, why is it that the light turns off causes a small step down in the conductivity? It seems that it's also generating some short lived carriers?
-
and the red lines represent the recombination paths, with solidand dashed lines representing large and small probability events, respectively.
Why are these different? Why is it much harder for a negative polaron to move out of it's well, than the positive polaron? (the dashed line in the upper right of the plot. Also, why is the negative polaron a deeper well thjan the positive polaron?
-
Difference between conventional and ferroelectric large polarons.
What are the red arrows, and why are there two charges in the polaron? Wikipedia says its a lattice distortion associated with a single charge.
-
ferroelectric nanodomains at low temperature,
Why not at high temperature?
-
linear dielectric to paraelectric and relaxor ferroelectric under increasingillumination.
What does that mean?
-
ray diffraction studiesreveal that photocarrier-induced structural polarization is present up toa critical freezing temperature
What's the critical temperature?
-
ultralong photocarrier lifetime beyond 106 s. X
What, so 11 days?
-
-
en.wikipedia.org en.wikipedia.org
-
when close to the phase transition temperature
transition to what phase?
-
-
-
Mach–Zehnder intensity modulator is used to generate psoptical pulses of SLED
What's that?
-
We use a 1550 nm laser diode(SFL1550S, Thorlabs) and a super luminescent light emitting diode(SLED; DL-CS5169A, Denslight) for the generation of ps opticalpulses.
Interesting. So they tried both sources.
-
in the pulsed Sagnac interfer-ometer, lasers with long coherence lengths perform well
They mean that it would have been long coherence, had it not been pulsed?
-
oblique-incidence designenables a study for longitudinal and transverse Kerr effects
How does that work?
-
all-fiber design simplifies optical alignment;17
still a a pain. Do they have a better approach?
-
s low noise and drift-free measurement
Ours drifts?
-
, limited by thephotodetector noise
How do they know that? Is that true for us?
-
superconducting states can be broken even with smalloptical powers.
interesting.
-
ejects all the reciprocal effects, such as linear birefringence andoptical activity unlike conventional MOKE microscopy
Oh?? Why does conventional MOKE see these?
-
100 nrad/√Hz sensitivityusing only a 10 μW optical power without the magnetic field modu-lation.
How do we compare with this?
-
ime-reversal symmetry breaking (TRSB) in condensed mat-ter systems generally results in magnetism
Wait what. Is that a causation?
-
1 μrad/√Hz sensitivityat a 3 μW
How does this compare with ours? And I presume the 3 uW is at the detector?
-
s the favorable properties of a Sagnac interferometer, such as rejection of all the reciprocal effects, a
Where is the proof of this?
-
- Oct 2023
-
en.wikipedia.org en.wikipedia.org
-
onceptual symmetries of resistor, capacitor, inductor, and memristor
Why is dphi/dt = V?
-
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Local file Local file
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Continuous” means that for every x ∈ M there exists a localparametrization ψ : U → RN of M at x with the property that Dψ(t) : Rn → Tψ(t) Mpreserves the orientation for all t ∈ U.
Mobius strip would satisfy this. However, it breaks down globally.
-
bj ∑ni1 a′i, j b′
Why do the subscripts of a appear backwards?
-
consisting of a frame(b1, b2, . . . , bn ) together with a sign ε ±1.
isn't this redundant. Just require that one pick a frame that's positive.
-
uniqueness is proved by verifying that the formula holds,
why does that prove uniqueness? (but not existence?)
-
are orthogonalvectors.
requires an inner product space.
-
gJ (x) ∑I φ∗ ( fI )(x) det(DφI,J (x)).
Unclear to me what this means exactly.
-
For ak-form α ∈ Ωk (V) define the pullback φ∗ (α) ∈ Ωk (U) by
Is this really a deffinition, or should it be derived? Because a pullback should change between coordinates in a meaningful way. Like we shouldn't have the freedom to define this, because it must be consistent in some way with natural law.
-
all k-multilinear functions of the formdxI dxi1 dxi2 · · · dxik
makes me think about tensor product vs wedge
-
β(bI )
Why is this true?
-
increasing multi-indices of degree k
Ah, here it must be increasing. (to answer the previous question)
-
multi-indices I and J l
Increasing or not?
-
There is a nice way to construct a basis of the vector space Ak (V) starting froma basis {b1, . . . , bn } of V
Seems like this is the answer to your previous question on whether it spans.
-
alternating k-multilinear functions is denoted by Ak (V)
Is this different from \(\Omega^k(V)\), the space of k-forms on V?
-
useful trick for producing alternating k-multilinear functions startingfrom k covectors µ1, µ2, . . . , µk ∈ V∗
Is the space of "alternating k-multilinear functions" spanned by the wedge product, or just that the wedge product maps into this space?
-
µ1µ2 · · · µk (v1 , v2 , . . . , vk ) det(µi (vj ))1≤i, j≤
What is the meaning of this definition?
-
which is simply the Jacobi matrix D g of g! (This is the reason that many authorsuse the notation dg for the Jacobi matrix.
For it to be a jacobi matrix, Dg would have to work for g multi dimensional.
-
Using this formalism we can write for any smooth function g on U
I get that that's true, but I don't see how this follows.
-
-
physics.stackexchange.com physics.stackexchange.com
-
A×B=A⊕BA×B=A⊕BA\times B=A\oplus B, but in the case of the product/sum of infinitely many vector spaces they are distinct: ∏iAi≠⨁iAi∏iAi≠⨁iAi\prod_i A_i\neq \bigoplus_i A_i. This wouldn't be something covered in introductory classes. The deep distinction between the two is that one is a category theory product and one is a category theory coproduct
Wild. I'd like to know what this means.
e.g. product vs coproduct and Cartesian product vs direct sum.
-
-
en.wikipedia.org en.wikipedia.org
-
such as having an odd number of electrons per unit cell.
Why does that make it a conductor?
-
-
Local file Local file
-
Edelstein effect
?
-
chirality represented by a pair of oppositely polarized spins
What does that mean?
-
applications in the chiral spintronics2 field.
What are these?
-
-
-
that when electrochem-ical water splitting occurs with an anode that accepts prefer-entially one spin owing to CISS, the process is enhanced andthe formation of hydrogen peroxide is diminished. [4
Why is that? Hydrogen peroxide isn't chiral is it?
-
Two recent examples area spin filter[39] and the emergence of a Hall voltage owing tospin accumulation. [24,34]
Look at thsese ones.
-
Figure 2.
why does positive voltage and negative voltage have the same effect? Considering that these systems are not symmetric to that flip, because the magnetic tip is only on one side of the sample.
Also, why does graph c which is supposedly non chiral show a splitting effect still? Is that jsut assumed to be noise?
-
chiralmaterials. [8,19,28–31]
weren't the previously mentioned items also chiral? like isn't this redundant?
-
CISS is repeatedly found tocorrelate with optical activity [28,33,3
What counts as optical activity.
-
Third, as in CISS in transmission, alsohere the sign of the preferred spin depends on the direction ofthe molecular dipole,
Is this what breaks the symmetry that allows you to select spin?
-
-
www.britannica.com www.britannica.com
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t-handed configuration. However, when these crystals were separated manually, he found that they exhibited right and left asymmetry
how do you separate the crystals manually?
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Louis Pasteur was the first to recognize that optical activity arises from the dissymmetric arrangement of atoms in the crystalline structures or in individual molecules of certain compounds.
How did he know that?
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