Is human vision sensitive to frequency or wavelength?

Is human vision sensitive to frequency or wavelength?

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In a vacuum, there is a one-to-one correspondence between light frequency ($ u$) and wavelength ($lambda$), ie. $lambda=c/ u$. But in a refractive medium, $lambda=v/ u$, so while the frequency may remain constant, the wavelength may not.

I've seen that in biology, wavelength is mostly used. But, I recall reading that human vision is sensitive to frequency. So, is human vision sensitive to wavelength or frequency?

It is not the wavelength or frequency that determines light absorption- it is the energy of the photon that matters. The energy of incident light should match the excitation energy of the chromophore. The medium itself can also affect light absorption by electronically interacting with the chromophore. Frequency is proportional to the energy (given by the relationship $E=h u$); while wavelength and velocity change in different media, frequency and energy remain constant.

The chromophore in the opsins (retinal) absorb energy to become active and you can say that they detect the frequency. However, sufficient number of opsins have to become active for the retinal cells to transmit the signal. So the overall light flux is also important. An interesting thing to note is that activation of the opsins hyperpolarizes the photoreceptor cell and reduces its tendency to fire.

Wavelength plays a role in diffraction and therefore diffraction is less in a medium with higher refractive index. Wavelength also determines scattering. So, while the opsins actually detect frequency, the wavelength can slightly affect the overall "vision" because of scattering/diffraction effects.

Wavelength-frequency relation is $lambda=frac{c}{ u}$ or $ ucdot lambda=c$ in vacuum. In some refractive media you can substitute $c$ with speed of light at corresponding medium, essentially nothing changes. Speed determines the product of wavelength and frequency, not their ratio, as you suggested. Hence at different speeds frequency and wavelength will both decrease or increase, there is no reason to think that one of them stays constant and the other one changes.

If you still want to make a distinction and describe physical property of light with one of them, frequency $ u$ would be more relevant as matter/light (photon/electron) interactions work as described by the formula $E=hcdot u$

As WYSIWYG wrote, energy is the answer, but to clarify the dependence of energy from momentum… sorry, the dependence of frequency from wavelength, have a look at the animation on this web page:, posted here in what I believe complies with fair use legislation:

It represents a sinusoidal wave, the basic component of any disturbance (as long as you can count on linearity, but that's another matter… ).

Frequency tells you how fast the red dot goes up and down. This is what a receptor put in that specific position will sense: a disturbance (here it's the height of a string, in your case the amplitude of the electric field) changing $ u$ times per second.

Wavelength, on the other hand, tells you how far the maxima and minima of the sinusoid are separated in the medium. In this case, two subsequent crests are separated by $lambda$ meters. For a given disturbance frequency, this number depends on how fast the wave is allowed to travel in the medium. A slow wave will have a shorter wavelength (because it will travel a smaller distance in space during the time the red dot completes a cycle); conversely, a fast wave will have a longer wavelength.

Note: the speed can depend on the frequency of the disturbance - in that case the medium is said to be 'dispersive' and the relation between frequency and wavelength (or energy and momentum, or - if you are into quantum physics - between energy E and wavenumber k) is called a 'dispersion relation'.

In an isotropic homogeneous medium, the speed is the same in every point and in any direction and you can write

$$lambda( u) = c( u) / u $$

The frequency is determined by the physical process in the source. It's very hard to change the frequency (i.e. the color, in the visible) of a light wave - you have to resort to nonlinear effects to do that. The wavelength is the result of the interaction with the medium, and changes all the time. But it's customary to express a sort of implied equivalence between frequency and wavelength based on the behavior the light wave would have in vacuum. Hence, when one says 550 nm photon, he is usually implying a photon with en energy corresponding to

$$E = h u = h c_0 /lambda = 4.226 10^-19 J = 2.64 eV$$

Where $c_0$ is the speed of light in vacuo. (Note: the latter value is computed from the one before diving by the charge of the electron).

In the context of human vision, wavelength and frequency are not mutually exclusive, but given by the first equation you pointed out in the question. That is, whenever the range of human vision is given they are not assuming that wavelength and frequency can change independently.

The sensitivity of the human eye to light of a certain intensity varies strongly over the wavelength range between 380 and 800 nm. Under daylight conditions, the average normal sighted human eye is most sensitive at a wavelength of 555 nm, resulting in the fact that green light at this wavelength produces the impression of highest "brightness" when compared to light at other wavelengths. The spectral sensitivity function of the average human eye under daylight conditions (photopic vision) is defined by the CIE spectral luminous efficiency function V( λ ) . Only in very rare cases, the spectral sensitivity of the human eye under dark adapted conditions (scotopic vision), defined by the spectral luminous efficiency function V'(ë), becomes technically relevant. By convention, these sensitivity functions are normalized to a value of 1 in their maximum.

As an example, the photopic sensitivity of the human eye to monochromatic light at 490 nm amounts to 20% of its sensitivity at 555 nm. As a consequence, when a source of monochromatic light at 490 nm emits five times as much power (expressed in watts) than an otherwise identical source of monochromatic light at 555 nm, both sources produce the impression of same "brightness" to the human eye.

Fig. II.13. Spectral luminous efficiency functions V( λ ) for photopic vision and V'( λ ) for scotopic vision, as defined by the CIE.

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      Is human vision sensitive to frequency or wavelength? - Biology

      Human vision is a complex process that is not yet completely understood, despite hundreds of years of study and research. The complex physical process of visualizing something involves the nearly simultaneous interaction of the eyes and the brain through a network of neurons, receptors, and other specialized cells.

      The human eye is equipped with a variety of optical elements including the cornea, iris, pupil, a variable-focus lens, and the retina, as illustrated above in Figure 1. Together, these elements work to form images of the objects in a person's field of view. When an object is observed, it is first focused through the cornea and lens onto the retina, a multilayered membrane that contains millions of light-sensitive cells that detect the image and translate it into a series of electrical signals. These image capturing receptors of the retina are termed rods and cones , and are connected with the fibers of the optic nerve bundle through a series of specialized cells that coordinate the transmission of the electrical signals to the brain. In the brain, the optic nerves from both eyes join at the optic chiasma where information from their retinas is correlated. The visual information is then processed through several steps, eventually arriving at the visual cortex, which is located on the lower rear section of each half of the cerebrum.

      Interactive Tutorial
      Human Vision Explore how images are produced on the retina of the human eye as an object is moved closer or farther away.

      A particularly specialized component of the eye is the fovea centralis , which is located on the optical axis of the eye in an area near the center of the retina. This area exclusively contains high-density tightly packed cone cells and is the area of sharpest vision. The density level of cone cells decreases outside of the fovea centralis and the ratio of rod cells to cone cells gradually increases. At the periphery of the retina, the total number of both types of light receptors decreases substantially, causing a dramatic loss of visual sensitivity at the retinal borders. This is offset, however, by the fact that humans constantly scan objects in their field of view, usually resulting in a perceived image that is uniformly sharp. A graphical illustration of the spatial arrangement of rod and cone cells and their connection to neurons within the retina can be seen below in Figure 2.

      Rod cells are most sensitive to green wavelengths of light (about 550-555 nanometers), although they display a broad range of response throughout the visible spectrum. They are the most populous visual receptor cells in humans, each eye containing about 130 million rods. Extremely responsive, the light sensitivity of rod cells is about 1000 times that of cone cells. However, the images generated by rod stimulation alone are relatively unsharp and confined to shades of gray, similar to those found in a black and white soft-focus photographic image. Rod vision is commonly referred to as scotopic or twilight vision because in low light levels it enables individuals to distinguish shapes and the relative brightness of objects, but not their colors.

      Cones, on the other hand, consist of three different types of cells, each "tuned" to a distinct wavelength peak of response centered at either 430, 535, or 590 nanometers. Often referred to as photopic vision, cone vision is dominant at normal light levels, both indoors and out. The quantity of cone cells possessed by humans, however, is much smaller than the number of rod cells, each eye only containing about 7 million cones. Stimulation of these visual receptors results in what is known as true color vision.

      The relative intensity of the stimulation incurred by each of the three types of cone receptors is what largely determines which color is imaged. For example, a beam of light that contains mostly blue short-wavelength radiation stimulates the cone cells that respond to 430-nanometer light far more than the other two cone types and, therefore, that light is seen as blue. Correspondingly, light with a majority of wavelengths centered around 550 nanometers appears green, and a beam containing mostly 600 nanometer wavelengths or longer is seen as red. When all three types of cone cells are stimulated equally, light is perceived as being achromatic or white. For instance, noon sunlight appears to humans as white light because it contains approximately equal amounts of red, green, and blue light, uniformly stimulating all types of cone receptors.

      Normal cones and pigment sensitivity enable humans to distinguish all of the different colors as well as subtle mixtures of hues. This type of color vision is known as trichromacy and relies upon the mutual interaction of all three types of photoreceptor cones. Yet, human color perception is also dependent upon illumination levels, shifts in color sensitivity occurring whenever lighting is varied. For example, blue colors look relatively brighter in dim light, while red colors appear more vivid in bright light. This effect can be simply observed by pointing a flashlight onto a color print, which results in the reds suddenly appearing much brighter and more saturated.

      Focus in the eye is controlled by a combination of elements, including the iris, lens, cornea, and muscle tissue. Properly functioning together, these components can alter the shape of the lens so that the eye can focus on both nearby and distant objects. However, in some instances the components do not work correctly or the eye is slightly altered in shape and the focal point does not intersect with the retina. As people age, for instance, the lenses of their eyes become harder and cannot be focused properly, which results in poor vision. If the point of an eye's focus is short of the retina, as illustrated in the lower part of Figure 3, the condition is called nearsightedness or myopia . People with this affliction are unable to focus on distant objects. In cases where the eye's focal point is behind the retina, as illustrated in the upper part of Figure 3, people have trouble focusing on nearby objects, which is a condition called hypermetropia , commonly known as farsightedness. These malfunctions of the eye can usually be corrected through the use of glasses, with concave lenses correcting myopia and convex lenses rectifying hypermetropia.

      Mortimer Abramowitz - Olympus America, Inc., Two Corporate Center Drive., Melville, New York, 11747.

      Shannon H. Neaves and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.

      The Human Eye, Optimized For Sunlight. Maybe.

      The human eye is sensitive to a portion of the electromagnetic spectrum that we call visible light, which extends from around 400 to 700 nanometer wavelength, peaking in the general vicinity of greenish light at 560 nanometers:

      Here's the intensity (formally: power per area per unit solid angle per unit wavelength - whew!) of the radiation emitted by an object with the temperature of the sun, plotted as a function of wavelength in nanometers according to Planck's law:

      Spectral radiance (W/(sr m^3)) vs. wavelength (nm)

      You'll notice it also peaks around the same place as the spectral response of the human eye. Optimization!

      Or is it? That previous equation was how much light the sun dumps out per nanometer of bandwidth at a given wavelength. But nothing stops us from plotting Planck's law in terms of the frequency of the light:

      Spectral radiance (W/(sr m^2 Hz)) vs. frequency (Hz)

      In this case what's on the y axis is power per area per unit solid angle per frequency. Ok, great. But notice it's not just the previous graph with f given by c/λ. It's a different graph, with different units. To see the difference, let's see this radiance per frequency graph with the x-axis labeled in terms of wavelength:

      Spectral radiance (W/(sr m^2 Hz)) vs. wavelength (nm)

      Well. This is manifestly not the same graph as the radiance per nanometer. Its peak is lower, in the near infrared and outside the sensitivity curve of the human eye. This makes some sense - there's not much frequency difference between light with wavelength of 1 kilometer and light with wavelength of 1 kilometer + 1 nanometer. But light of 100 nanometer wavelength has a frequency about 3 x 10 13 Hz more than light with wavelength 101 nanometers.

      So what gives? Is the eye most sensitive where the sun emits the most light or not? The simple fact of the matter is there's no such thing as an equation that just gives "how much light the sun puts out at a given wavelength". That's simply not a well-defined quantity. What is well defined is how much light the sun puts out per nanometer or per hertz. In this sense our eye isn't optimized so that its response peak matches the sun's emission peak, because "the sun's peak" isn't really a coherent concept. The sensitivity of our eyes is probably more strongly determined by the available chemistry - long-wavelength infrared light doesn't have the energy to excite most molecular energy levels, and short-wavelength ultraviolet light is energetic enough to risk destroying the photosensitive molecules completely.

      This wavelength/frequency distribution function issue isn't just a trivial point - it's one of those things that actually gets physicists in trouble when they forget that one isn't the same thing as the other. For a detailed discussion, I can't think of a better one than this AJP article by Soffer and Lynch. Enjoy, and be careful out there with your units!

      More like this

      I have a favourite graph from a physics textbook. It's really wonderful. Jackson's Classical Electrodynamics, the second edition. Page 291, Figure 7.9. The absorption coefficient for liquid water as a function of linear frequency. It's quite striking. EM radiation at 1e+14 Hz has an absorption coefficient of 1e4 inverse cm. Then, over less than two decades of frequency, the absorption plummets almost 8 decades to the middle of the visible spectrum. Then, over the next decade in frequency again, the absorption climbs by 10 decades. Because our eyes contain a large amount of liquid water, simple intraocular absorption limits us to a frequency range between about 2e+14 and 1.1e+15 Hz. If the absorption coefficient is more than about 3 or 4 inverse cm, there will be high attenuation between the lens and the retina.

      Some insect eyes don't have the large amount of free water inside them, and have a shorter distance from outside the eye to the imaging surface, and so can adapt to a wider frequency range, but when your eye is a bag of mostly water the size of a grape, it'll have a frequency response not terribly far from that of humans.

      It's also worth noting that when you're talking about "most of the light" you're really talking about is "most of the light energy." You can also make a case, even though our senses are tuned to quantity of energy, that "amount of light" is better thought of as "amount of photons." Then you get two more peaks to play with. :)

      Another plausible curve is energy vs the logarithm of wavelength (or frequency, other than a sign flip it won't make any difference), which peaks at around 610 nm. You can also the wavelength where 50% of the energy is above it (about 710) or the wavelength where 50% of the photons are shorter (don't have that on hand).

      However, it's certainly not most of the light, since 400-700 nm is only about 37% of the light.

      Matt Springer: "So what gives? Is the eye most sensitive where the sun emits the most light or not?"

      It isn't at all clear why it would be necessary to have the sensitivity peak at the same place.

      Put another way, it might be more useful to have the sensitivity peak somewhere other than where the sun emits the most light.

      A couple of colleagues of mine recently wrote a paper on exactly this subject, arguing for a different pedagogical presentation of blackbody radiation. I haven't read it all that carefully, but it seems worth plugging here.

      I like it. It makes much more sense than most presentations.

      My only reservation is my own Willis Lamb style prejudices about "number of photons emitted per second", because "number of photons" is not well-defined for classical thermal radiation.

      In reply to by Chad Orzel (not verified)

      Similar to what chad's colleagues suggest wouldn't it be more appropriate, in the case of eye development, to compare the photon flux of relevant light sources (in this case the sun) with the energy dependent quantum efficiency of the eye? Anyone knows how that would look?

      I should check it myself, but the one-year-old pulling in my arm thinks otherwise, so I'm just throwing out the suggestion.

      I believe I missed something in your reparameterization of B(T). Why does one version have lambda^-5 while the other has nu^3? This does not work if lambda*nu = c.

      Nevermind, the wikipedia page shows the differential relation for the reparameterization.


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      You can also shop using Amazon Smile and though you pay nothing more we get a tiny something.

      Let me try to get this thread going.

      Humans sense external heat sources, sunlight and re-radiated longer wavelengths, ultimately through sensory neurons -

      is vague to me. What you are really asking is: what is the absorption spectrum (I think) for human skin tissue surrounding Thermoreceptors?
      doi:10.1117/1.JBO.17.9.090901 Optical properties of human skin man-skin/10.1117/1.JBO.17.9.090901.full?SSO=1

      Hemoglobin and melanin are most active molecules in skin -- for absorbing light and therefore generating a sensory response. The link I provided seems good but is technical. Look for graphs, those are easy to understand.

      Thank you jim mcmanara. I have read the article and looked at the graphs. Very interesting. That said, the graphs go up to 750nm which is near infrared (NIR).

      I am still wondering how FIR, i.e. wavelengths longer than 2.5 micron, are absorbed by the skin and makes us feel hot or if they just penetrate below the epidermis. Most objects,under sunlight get very hot and their temperatures are such that their emission peak is in the 10micron or so. the human body emits strongly at 9 micron. Does that mean that is also efficiently and strongly absorbs energy at that same wavelength due to resonance?

      We humans are homeothermic, we regulate our body temperature. We do this in several ways:
      constrict epidermal blood flow to conserve body heat in low ambient temperature,
      increase epidermal blood flow to remove heat via sweat and direct radiation in hot ambient temperature.

      You are making more of this than it is worth. Yes, we absorb and re-radiate heat (why some mammal eating reptiles have Jacobsen's organs to detect heat signatures of mammalian prey).

      Feeling hot or cold is a brain/neuron function. Nothing else.

      If the entire human body throughout was at ambient temperature, humans would die off pretty quickly almost eveywhere. Humans start to lose consciousness when internal body temperatures are less than 94°F or greater than

      108°F. Long periods of time in vast areas of Earth experience temperatures outside that "golden zone".

      You may be right. I guess I am just wondering, from a purely physics standpoint, which, between far infrared or near infrared radiation, in the same amount, is more effective at keeping as warm.

      You may be right. I guess I am just wondering, from a purely physics standpoint, which, between far infrared or near infrared radiation, in the same amount, is more effective at keeping as warm.

      It depends on how much of each wavelength your body/clothes absorb. If you absorb 700 nm light better than 2500 nm then the former would be more efficient at keeping you warm as long as the incoming energy at each wavelength is the same. However when dealing with sunlight there is more energy in the near IR band than the FIR band, so it likely plays a much larger role in warming you during the day.

      Note that 'keeping us warm' is not the same as 'feels warm'. If lots of radiation is absorbed near your thermoreceptors, you may initially feel warmer than if the radiation was of a wavelength that was absorbed deeper in your body where you have little thermoreceptors. For example, microwaves and radio waves can penetrate your skin and heat your internal organs directly, without you ever feeling very warm. Absorbing 100 watts of IR radiation would heat you less than 500 watts of microwave radiation, but you may feel the former on your skin more than the latter.

      But I would say that shorter wavelength radiation, like UV and visible, are not very penetrating and get absorbed on the outer skin layers and produce the feeling of being warm. IR goes deeper and may still cause heating. But when we are outside under sunlight, we probably feel hot mainly because of the UV and visible.

      Is human vision sensitive to frequency or wavelength? - Biology

      Spectral sensitivity is the relative efficiency of detection, of light or other signal, as a function of the frequency or wavelength of the signal.

      In visual neuroscience, spectral sensitivity is used to describe the different characteristics of the photopigments in the rod cells and cone cells in the retina of the eye. It is known that the rod cells are more suited to scotopic vision and cone cells to photopic vision, and that they differ in their sensitivity to different wavelengths of light. It has been established that the maximum spectral sensitivity of the human eye under daylight conditions is at a wavelength of 555 nm, while at night the peak shifts to 507 nm.

      In photography, film and sensors are often described in terms of their spectral sensitivity, to supplement their characteristic curves that describe their responsivity. A database of camera spectral sensitivity is created and its space analyzed. For X-ray films, the spectral sensitivity is chosen to be appropriate to the phosphors that respond to X-rays, rather than being related to human vision.

      In sensor systems, where the output is easily quantified, the responsivity can be extended to be wavelength dependent, incorporating the spectral sensitivity. When the sensor system is linear, its spectral sensitivity and spectral responsivity can both be decomposed with similar basis functions. When a system’s responsivity is a fixed monotonic nonlinear function, thatnonlinearity can be estimated and corrected for, to determine the spectral sensitivity from spectral input–output data via standard linear methods.

      The responses of the rod and cone cells of the retina, however, have a very context-dependent (coupled) nonlinear response, which complicates the analysis of their spectral sensitivities from experimental data. In spite of these complexities, however, the conversion of light energy spectra to the effective stimulus, the excitation of the photopigment, is quite linear, and linear characterizations such as spectral sensitivity are therefore quite useful in describing many properties of color vision.

      Spectral sensitivity is sometimes expressed as a quantum efficiency, that is, as probability of getting a quantum reaction, such as a captured electron, to a quantum of light, as a function of wavelength. In other contexts, the spectral sensitivity is expressed as the relative response per light energy, rather than per quantum, normalized to a peak value of 1, and a quantum efficiency is used to calibrate the sensitivity at that peak wavelength. In some linear applications, the spectral sensitivity may be expressed as a spectral responsivity, with units such as amperes per watt.

      Infrared Waves and Eye Damage

      People who work in industries which expose them to infrared radiation for long periods of time may experience eye damage. The human eye is sensitive to all of the radiation in the electromagnetic spectrum, especially if that radiation is at very high levels of intensity. Exposure to intense electromagnetic radiation, including infrared radiation, can damage the lens and cornea of the eye. This is one reason why staring at the sun is harmful (and unintelligent). People who work near intense radiation must wear goggles.

      Visualizing color spaces & chromaticity

      So far most of our graphs have put wavelength on the horizontal axis, and we’ve plotted multiple series to represent the other values of interest.

      Instead, we could plot color as a function of $(R, G, B)$ or $(L, M, S)$. Let’s see what color plotted in 3D $(R, G, B)$ space looks like.

      Cool! This gives us a visualization of a broader set of colors, not just the spectral colors of the rainbow.

      A simple way to reduce this down to two dimensions would be to have a separate plot for each pair of values, like so:

      Component pairs plotted, holding the third coordinate constant

      In each of these plots, we discard one dimension by holding one thing constant. Rather than holding one of red, green, and blue constant, it would be really nice to have a plot showing all the colors of the rainbow & their combinations, while holding lightness constant.

      Looking at the cube pictures again, we can see that (0, 0, 0) is black, and (1, 1, 1) is white.

      What happens if we slice the cube diagonally across the plane containing $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$?

      This triangle slice of the cube has the property that $R + G + B = 1$, and we can use $R + G + B$ as a crude approximation of lightness. If we take a top-down view of this triangular slice, then we get this:

      This two dimensional representation of color is called chromaticity. This particular kind is called rg chromaticity. Chromaticity gives us information about the ratio of the primary colors independent of the lightness.

      This means we can have the same chromaticity at many different intensities.

      We can even make a chromaticity graph where the intensity varies with r & g in order to maximize intensity while preserving the ratio between $R$, $G$, and $B$.

      Chromaticity is a useful property of a color to consider because it stays constant as the intensity of a light source changes, so long as the light source retains the same spectral distribution. As you change the brightness of your screen, chromaticity is the thing that stays constant!

      There are many different ways of dividing chromaticity into two dimensions. One of the common methods is used in both the HSL and HSV color spaces. Both color spaces split chromaticity into “hue” and “saturation”, like so:

      It might appear at a glance that the rg chromaticity triangle and these hue vs. saturation squares contains every color of the rainbow. It’s time to revisit those pesky negative values in our color matching functions.

      Assuming human skin is at 98.6 degrees Fahrenheit, what wavelength is the peak in the human thermal radiation spectrum? What type of waves are these?

      #lambda_(max)=9.34 mu m# which is called infrared light.


      Wien's displacement law states that the black-body radiation curve for different temperatures peaks at a wavelength inversely proportional to the temperature.

      where b is Wien's displacement constant, equal to #2.898×10^(−3) m K# and #T# is the temperature in Kelvin. The temperature we were given needs to be converted to Kelvin:

      #T = (T_F-32^oF)*(5K)/(1^oF) + 273K = 310.15K#

      plugging this into our equation, we get the peak wavelength of radiated light:

      #lambda_(max)=(2.898×10^(−3) m K)/(310.15K)=9.34x10^-6m=9.34 mu m#

      This is in the range of what is called infrared radiation.

      Should be the infrared fingerprint region.

      • #b = 2.89777xx10^(-3)# #"m"cdot"K"# is a proportionality constant, probably experimentally determined.
      • #T# is temperature in #"K"# .
      • #lambda_max# is the wavelength that you observe at its largest spectral energy density.

      The spectral energy density is depicted in the following diagram, with respect to wavelength in #"nm"# :

      You can think of the spectral energy density as being proportional to the contribution of each wavelength range to some final observed color at a particular temperature. You can see that the peaks would correspond to #lambda_max# .

      Converting temperature to #"K"# , we get:

      And now we get a max wavelength of:

      Converting this to #mu"m"# , we get:

      Being close to #10# #mu"m"# , I find that it's close to the infrared region. And if you convert to #"cm"^(-1)# , you should get #"1070.32 cm"^(-1)# , which is within the "fingerprint" region of the IR spectrum.


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