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Why not blue?

Posted by npthaskell coastal oregon (My Page) on
Mon, Jan 2, 06 at 0:10

Shrubs_n_bulbs, what exactly do you have against lamps that put out a lot of blue, such as those used in the aquarium world.

My lettuce and radish is getting a a bit leggy under GE's "plant & aquarium, wide spectrum" fluorescent. I've just switched to 6500K Philips "Daylight Delux", and am thinking of going towards 8,000 to 20,000 K aquarium fluorescents.


Follow-Up Postings:

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RE: Why not blue?

If your lettuce are getting leggy, its because the light isn't bright enough. Changing from your 6500K lamp to one with a different spectrum will not help. Add another 6500K lamp and get some sunglasses :)

The actinic and coral lamps are inefficient for producing plant growth (too little light altogether and too little red light), even potentially harmful because many of them produce too much UV light. If they were so great for growing plants, don't you think someone would have noticed and be selling them labelled as magic plant lamps? Why not try them on half your crop and report back to us?


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RE: Why not blue?

  • Posted by DanaNY z6 Brooklyn, NY (My Page) on
    Mon, Jan 2, 06 at 15:35

6500k should be perfect for lettuce. I grow lettuce under 6400k and they're not leggy. Just keep the tubes very close to the plants, no more than a couple inches. If the lights are too far away, they will stretch. Warm temps can also cause lettuce to get leggy.


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RE: Why not blue?

> If your lettuce are getting leggy, its because the light isn't bright enough.
> Changing from your 6500K lamp to one with a different spectrum will
> not help. Add another 6500K lamp and get some sunglasses :)

Suppose that the light intensity (as measured by photon flux) is at the borderline between "enough" and "not enough". Under such conditions, I assume that increasing the blue to red ratio would be beneficial in reducing leggyness.

> The actinic and coral lamps are inefficient for producing plant growth
> (too little light altogether and too little red light)...

Perhaps you have seen plots of photosynthesis as a function of wavelength, in which blue has a effectiveness of ~ 1/2, green ~ 2/3 (green would be about ~3/4, but has a little dip in the plot), and red ~1 (where by definition, I normalized the effectiveness of red to "1"). Such plots result when the light intensity is measured in watts per square meter. When light intensity is measured by photons per square meter (microEinstiens), the plots show that blue and red are both ~1, and the dip in the green is more pronounced. (The dip in green is more obvious with pale thin leaves or dilute algae, as compared to plants with dark thick leaves.) In conclusion, the claim that blue light is less effective than red light in driving photosynthesis is an illusion and not a real issue.

Plants have a blue sensor "cryptochrome" and a red sensor "phytochrome". The first real issue is whether growing plants under aquarium lights results in a harmful imbalance in the cryptochrome versus phytochrome signals. Actually I am hoping for an imbalance, an imbalance that retards bolting!

The second real issue is whether a red lamp can put out twice as many photons as a blue lamp (given equal consumption of watts). This is a question of economics and enviornmental stewardship. I don't believe that a hypothetical fluorescent tube coated with only red phosphor emits more photons than one coated with only blue phosphor. One would think that if a photon of UV (emitted by mercury excited by electric current), absorbed by a blue phosphor, is converted into one photon of blue; then a photon of UV absorbed by a red phosphor would be converted into two photons of red. But this is not the case, the currently known phosphors are not that "intelligent", and so much of the UV energy that could be used is dumped as heat. It gets even worse in the case of non-hypothetical white lamps with a mixture of phosphors. Either the green or the red phosphor (I forget which) doesn't absorb UV well, while the blue phosphor does. Either the green or red phosphor has to absorb blue photons emitted by the blue phosphor, and less than 100% of the blue photons absorbed by the green or red phosphors succeed in being converted into green or red. Those blue photons that fail are converted into waste heat. For each color of light (R,G,B) there are about 3-5 several commercially available phosphor alternatives. There may be some white combinations that do not require the design of a phosphor to phosphor relay system (all phosphors may absorb UV directly at high efficiency). But in such a phosphor mix, there is a phosphor to phosphor relay by accident, where green and red emission gets short changed because the relay is not 100% efficient. I conclude that blueish fluorescent lamps could be more efficient than redish lamps, given current phosphors. The realization of this potential depends on which of the commercial blue phosphors is actually used.

>potentially harmful because many of them produce too much UV light.

Possibly true for some aquarium metal halides. But the UV to blue ratio emitted by most fluourescents is lower than that of sunlight itself. This is even true for some full spectrum bulbs with UV emitting phosphors sold for people with seasonal affective disorder, and aquarium actinic bulbs with a peak at 420nm (the UV output of the 420nm phosphor decays much faster than the filtration of sunlight by the atmosphere). Of course, blacklights, tanning lamps, and some reptile lamps may be problematic. In regards to the UV component of aquarium fluorescents and metal halides, I worry much more about UV-induced degradation of household paint and fabrics than about inhibition of plants.

> If they were so great for growing plants, don't you think someone
> would have noticed and be selling them labelled as magic plant lamps?

Well, they are sold as magic lamps! Magic lamps to drive photosynthesis in the dinoflagellate symbionts found in coral. Plant lamps are "magic" if they enable plants to flower; dinoglagellates don't "flower". I don't want my plants to flower! I have the impression that the phosphor in classic "actinic" aquarium lamps (those that emit blue at a 420nm peak) degrades quickly. A short lifespan may be the reason for such narrow marketing. Many alternative (newer?) aquarium lamps have at least one blue phosphor that peaks at ~450nm; I don't know how its lifespan compares.

> Why not try them on half your crop and report back to us?

My current outlay is too small; in a few months I'm moving and will have a larger area with a ceramic metal halide in addition to several fluorescents.

In a few hours I will post more about how aquarium lamps may relate to cryptochrome and phytochrome. In the meantime, here is an interesting link in regards to the use of filters in greenhouses to manipulate phytochrome

Here is a link that might be useful: Clemson color filters


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RE: Why not blue?

I wrote (partially corrected version below):

> For each color of light (R,G,B) there are about 3-5 commercially
> available phosphor alternatives.

Here are phosphors that emit ~450nm (Sylvania catalog #, chemical formula)

1A. 2196 BaMgAl10O17:Eu:Mn
1B. 2464 BaMgAl10O17:Eu
2A. 2412 CaWO4:Pb
3. 243 Sr2P2O7:Sn
4. 2471 Sr5Cl(PO4)3:Eu

Phosphors that emit ~480 (bluish-blue-green)

5. 2301 MgWO4
6. 242 (Ba,Ti)2P2O7:Ti
7. 248 Sr6P5BO20:Eu

Halophosphors with broad tunable emission

8A. 2440 Ca5F(PO4)3:Sb
8B. 4990 Ca5(F,Cl)(PO4)3: Sb:Mn

Actinic phosphor (~420)

2B. 2402 CaWO4:Pb

After actually counting, there are about 8-11 groups of blue phosphors alone (not 3-5). And each group has multiple variants that differ slightly in chemistry and physical processing. I doubt that this list includes the real aquarium "actinic" 420nm phosphor; it includes the closest match. I believe that all of these can be used in fluorescent lamps, except I have doubts about 2A & 2B (my memory is that these are X-ray phosphors).

Here is a link that might be useful: Phosphors from Sylvania


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RE: Why not blue?

I wrote:

> Either the green or the red phosphor (I forget which) doesn't absorb
> UV well, while the blue phosphor does....phosphor to phosphor
> relay system....I conclude that blueish fluorescent lamps could
> be more efficient than redish lamps....

This argument, that blue fluorescent lamps could be more efficient, was based on a faulty memory of an incorrect interpretation of the article linked below, which said,

"The Y2O3:Eu3+ phosphor [the orange/red component of triphosphor lamps] absorbs... peaks at about 230nm. Hence, the absorption of 254 nm
radiation is not very high with plaques of the commercial formulation reflecting about 25 percent of this radiation...The high reflectivity and the fact that the green and blue phosphors are stronger absorbers of the 254 nm radiation requires the Y2O3:Eu3+ phosphor to be the dominant component (by weight) of the triphosphor blend."

This quote does not imply that blue and green phosphors are required for the efficient transfer of energy from mercury excitation to the red phosphor. I mistakenly thought that it (along with other info) did.

Here is a link that might be useful: Fluorescent Lamp Phosphors (64Kb pdf)


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RE: Why not blue?

Suppose that the light intensity (as measured by photon flux) is at the borderline between "enough" and "not enough". Under such conditions, I assume that increasing the blue to red ratio would be beneficial in reducing leggyness.

Nope, it is a myth that internode length is increased by red light and decreased by blue light. Red light, or orange light such as HPS, produces very short internodes (ie. no legginess). The issue with such light is that they produce excessively branched and compact plants and biomass is not maximised on such small plants. There are also other biological problems with a complete lack of blue light. Etiolation is controlled very strongly by the ratio of red light to far red light, but this is rarely an issue with fluorescent lights because they just don't produce enough far red light. Incandescents do and they have even been used to supplement HPS lighting to stretch plants. One of the biggest problems with fluorescent lights is that the main red phosphors have a sharp dropoff in light at the most efficient wavelengths for photosynthesis. This is intentional since the human eye is not very sensitive to those wavelengths. The common blue phosphors do peak right at the best place for plants.

Perhaps you have seen plots of photosynthesis as a function of wavelength, in which blue has a effectiveness of ~ 1/2, green ~ 2/3 (green would be about ~3/4, but has a little dip in the plot), and red ~1 (where by definition, I normalized the effectiveness of red to "1"). Such plots result when the light intensity is measured in watts per square meter. When light intensity is measured by photons per square meter (microEinstiens), the plots show that blue and red are both ~1, and the dip in the green is more pronounced. (The dip in green is more obvious with pale thin leaves or dilute algae, as compared to plants with dark thick leaves.) In conclusion, the claim that blue light is less effective than red light in driving photosynthesis is an illusion and not a real issue.

You have just pointed out that a photosynthetic plot against power (watts) shows blue light to be half as effective as red light. Similar plots against photons (micro-einsteins) show that a blue photon is roughly as effective as a red photon, but unfortunately it takes more energy to produce a blue photon, hence a 40W fluorescent can produce less blue photons than red photons purely based on physics. Add in that blue phosphors are generally less efficient and degrade more quickly and you have a problem. Remember that you buy electricity and fluorescent tubes by the watt, not by the photon.

Actually I am hoping for an imbalance, an imbalance that retards bolting!

Now here you have a situation where blue light can be beneficial. Some plants bolt when grown with insufficient blue light. Approximately 10% blue light by photon count is sufficient to prevent this, which approximates to the 25% blue light by wattage in the good old warm white cool white combo. 6500K tubes actually produce more blue light than needed for this, but they are not wildly out of line and they are often made in "wide spectrum" forms that produce a good spread of light in the best red wavelengths. They also often have a high CRI for good colour rendition if you want your plants to look good under lights.

The second real issue is whether a red lamp can put out twice as many photons as a blue lamp (given equal consumption of watts). This is a question of economics and enviornmental stewardship. I don't believe that a hypothetical fluorescent tube coated with only red phosphor emits more photons than one coated with only blue phosphor.

Again, pure physics says that it requires more energy to produce a blue photon than a red one, proportionate to the frequency, or inversely proportional to the wavelength. Using blue and red phosphor peaks at 420nm and 650nm, you get a ratio of 64% of blue photons compared to red for the same power. Add in the difficulties with blue phosphors and a red phosphor most certainly does emit more photons per watt than a blue one (or even a green one). Remember that one blue photon achieves, at best, the same photosynthetic effect as one red photon.

In regards to the UV component of aquarium fluorescents and metal halides, I worry much more about UV-induced degradation of household paint and fabrics than about inhibition of plants.

Definitely. Metal halides degrade paint over time. Aquarium fluorescents may or may not produce enough UV to be harmful. Some, such as actinics, produce very little UV. Even some non-actinics produce just a narrow spread of blue and violet light, but others produce a wider spread including quite a bit of UV. Coral lamps in particular tend to produce a higher proportion of UV than is good for plants. Try a coral lamp on surface plants at an intensity comparable to sunlight and let us know what happens :) Maybe nothing, maybe crispy plants, but why take the risk when a saltwater aquarium lamp offers nothing extra for growing green plants.

Well, they are sold as magic lamps! Magic lamps to drive photosynthesis in the dinoflagellate symbionts found in coral.

Yes, that's right, they are designed for corals that live several feet under saltwater where they receive very little red light and probably wouldn't know what to do with a red photon if they tripped over it :) Coral lamps are actually designed to produce an excess of violet and UV light to enhance natural fluorescence and make the corals look nice.

I have the impression that the phosphor in classic "actinic" aquarium lamps (those that emit blue at a 420nm peak) degrades quickly.

Actinic lamps certainly don't have the same listed lifetimes as regular triphosphor fluorescents. I don't know how much of this is just lack of effort on the part of the manufacturers. I started a thread many months ago about apparent spectral changes in triphosphor fluorescents as they age since the blue triphosphor phosphors are the same as the phosphors in at least some of the aquarium lamps. It does seem that the blue phosphors degrade more quickly (and so a triphosphor tube becomes more red) but it is difficult to find hard data on this.

Your Clemson link is interesting. It describes the primary spectral effect on plant height, which is the ratio of far red light. They also mention right at the end the possible effects of far red light at particular times of the day. This has been studied elsewhere and it appears that relatively small amounts of far red light have a huge effect at particular times, especially right before or during the dark period.


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RE: Why not blue?

> Again, pure physics says that it requires more energy to produce a blue
> photon than a red one, proportionate to the frequency, or inversely
> proportional to the wavelength. Using blue and red phosphor peaks at
> 420nm and 650nm, you get a ratio of 64% of blue photons compared to
> red for the same power. Add in the difficulties with blue phosphors and a
> red phosphor most certainly does emit more photons per watt than a blue
> one (or even a green one).

In a fluorescent lamp, electrical energy excites mercury vapor, which mostly emits UV at 254nm (~4/5), hard UV at 174nm (~1/5), and spikes in the near UV and visible (~3%). The number of UV photons per watt emitted by mercury is a function of the design of the tube, ballast, mercury sorce, fill gas, etc. The identity of the phosphor mix doesn't feed back and tell mercury how efficient it should be in converting Watts into UV photons. I've read that the efficiency of conversion of electrical power into UV energy is about 50%; the other is dissapated as heat. I'm going to talk a lot about heat emissions after this step (heat emissions when UV is converted into Visible light, and when visible light is absorbed by chlorophyll). From now on, I'll ignore the heat dumped during the excitation of mercury. The conversion of UV into visible energy by white phosphor mix is again about 50%; half to light and half to heat. The total conversion of electrical power to white light is then about 25% (50% of 50%)

Commercial phosphors have quantum efficiencies near 100%, ie., above 80-90%. Since the number of UV photons generated by electrical current does not depend on the identity of the phosphor, then the number of red photons emitted (per watt of electric power) from a tube coated with only a red phosphor will equal the number of blue photons emitted per watt from a tube coated with only a blue phosphor.

A blue phosphor converts one UV photon to one blue photon plus a LITTLE heat. UV photons have more energy than blue, and the energy gap between the two is dumped as heat.

A green phosphor converts one UV photon to one green photon plus a MODEST amount of heat. The amount of dumped heat is equal to the energy gap between green and UV photons (this in turn is equal to the energy gap between green and blue photons, plus the energy gap between blue and UV photons).

A red phosphor converts one UV photon to one red photon plus a LOT of heat. The amount of dumped heat is equal to the energy gap between red and UV photons (this in turn is equal to the energy gap between red and green photons, plus the energy gap between green and blue photons, plus the energy gap between blue and UV photons.

A red photon [600-700nm] will be absorbed by a bed of antennae chlorophyll within the leaf, where it is converted into a far-red photon [680-700nm] (plus a little heat!) within the antennae bed, and then the far-red photon is delivered to reaction center chlorophyll, where it drives the photochemistry of photosynthesis.

A blue photon will be absorbed by a bed of antennae chlorophyll, where it is converted into a far-red photon (plus a lot of heat!!!), same steps as above.

In summary, from mercury vapor to plant, one UV photon (254nm) is emited and finally downconverted to one far-red photon (680-700nm); the energy gap between UV and far-red is dumped as heat. The total amount of dumped heat is the same whether a blue or red phosphor coats the lamp. The distribution of where the heat is actually dumped (lamp coating versus leaf interior) differs a little bit depending on which phosphor is used.

Commercial phosphors have quantum efficiencies near 100%, ie., around 80-95% . This small variation in efficiency isn't really a function of the color emission of the phosphors. The 8-11 commercial blue phosphors have efficiencies of 80-95%, as do the green and red phosphors. So we can say that a lamp coated with a red phosphor is equivalent to a lamp coated with blue phosphor, in driving photosynthesis per watt of electrical power, within + or - 15%.

In a white lamp, the red phosphor is driven by UV from mercury, plus recapture of some of the emission from the green phosphor, plus recapture of some of emission from the blue phosphor. The green phosphor is driven by UV plus recapture of some of the emission from the blue phosphor. The blue phosphor is only excited by UV. Of course, most of the green emission escapes recapture by the red phosphor; and most of the blue emission escapes recapature by the green and red phosphors. If the secondary emission from "recaptured primary emission" is about 90% efficient (on a photon basis), then the ability of a white lamp to drive photosynthesis per Watt could be about 10% less than that from the best pure red or pure blue lamps.

For plants with dark thick leaves, green light is about 80-90% as efficient in driving photosynthesis as compared to either blue or red (on a photon basis). For dilute algal suspensions or pale thin leaves, green light can be only 10% as efficient as blue or red. Considering that white lamps emit mostly green, and assuming dark thick leaves, perhaps an extra 10% should be subtracted from the efficiency of white lamps compared to pure red or blue.

If the efficiency of pure red or pure blue fluorescents, in terms of photosynthesis per watt of electrical power, is about the same; it follows that the efficiency of plant growth per Watt would be the same for these two hypothetical lamps. Such a conclusion is only true if pure blue or pure red lamps don't have hormonal effects that decouple plant growth from a direct linear relationship with photosynthesis.

I hope to address some of your points later, this post is long enough.


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RE: Why not blue?

Just three quick clarifications, before I address other issues in another long post...

> Again, pure physics says that it requires more energy to produce a blue
> photon than a red one, proportionate to the frequency, or inversely
> proportional to the wavelength.

Everything that I have posted is compatible with your statement here. It is precisely because it takes more energy to produce a blue photon that a blue phosphor can't afford to dump as much heat as a red phosphor does. Here are my calculations:

A blue phosphor converts about 56% of the energy of a UV photon (254nm) into a blue photon (~450nm), with 44% of the UV photon's energy being dumped as heat. A green phosphor converts about 46% of the energy of a UV photon into a green photon (~ 550nm) with 54% of the UV photon's energy being dumped as heat. A red phosphor converts about 39% of the energy of a UV photon into a red photon (~650nm), with 61% of the energy of a UV photon being dumped as heat.

--------

My argument, that a fluorescent lamp coated with only blue phosphor is as efficient as a lamp coated with only red phosphor, implicitly assumes the following: 1) The lamps are fresh; aged and degraded phosphors are a different game. 2) By "blue" or "red" phosphors, I mean that specific shades of blue or red emited by the phosphor match chlorophyll peaks, at least "close enough"; this assumption can be relaxed if the leaves are dark and thick.

-------

One of my reference links doesn't work, try:

Here is a link that might be useful: corrected - Fluorescent Lamp Phosphors (64Kb pdf)


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RE: Why not blue?

That's a great document. I've learned a lot. I've learned that the primary red phosphor is more efficient (and lasts longer) than the blue phosphors but not by as much as I thought and not for the reasons that I thought.

Still, even assuming the blue phosphors were absolutely as efficient as the red ones in generating photons, then these high colour temperature and actinic tubes are still expensive, don't last long, and give you a spectrum that isn't good for growing plants.


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RE: Why not blue?

Nice thread. It's good to see more folks discussing the phosphor aspect of lighting. Once you get the idea of what the different phosphors produce, you realize how common certain phosphors are among similar lamp types, produced by different manufacturers.

The IF6-03-Pages48-51.pdf is a great phosphor document. The other document definitely worth searching for is IF6-03-Pages48-51.pdf. Those two PDF files complement one another.

Tricolor phosphors(rare earth)are in almost ALL of the CFL's and in most of the newer linear fluorescents being produced these days. They also last longer than the older halophosphors, as you all mentioned. Because of that fact, some lamps, such as many of the Sylvania 70CRI series lamps, have the halophosphor applied first to the surface of the glass. Then they apply the tri-phosphor layer. That way, the plasma column doesn't wear out the halophosphor as soon as it would otherwise.

Unfortunately, as one of you mentioned, the common red phosphor in the tri-color scheme (Y2O3:Eu, Yttrium Oxide) is attuned to the human eye at 611nm. Not where we want it... and NARROW to boot!

The only other two fluorescent lamp phosphors that give better red are:
(Sr,Mg)3(PO4)2:Sn (Strontium magnesium phosphate)
This is the phosphor used to coat most of the "Deluxe" coated versions of HID lamps, mostly Mercury Vapor. It has a wideband and is centerd at about 622nm.
3.5MgO0.5MgF2GeO2:Mn (Magnesium fluorogermanate)
This is the phosphor used in the GRO-LUX lamps. It is centered at 658nm and is not as wide as the one above.

By the way. Most people don't realize that the term "phosphor" refers to the phosphorescent nature of the compounds... NOT because they contain any phosphate.

Zink


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RE: Why not blue? CORRECTION!!

CORRECTION!
One paragraph above should read:

"The IF6-98-Pages28-31.pdf is a great phosphor document. The other document definitely worth searching for is IF6-03-Pages48-51.pdf. Those two PDF files complement one another."

Zink


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Blue, Red, Biology, Phytochrome, Cryptochrome

Let me begin a new offshoot from the thread "Why not blue?". This new thread will discuss how lamps possessing different spectral outputs effect the biology of plants. The "why not blue?" thread is getting rather technical and non accessable to newbies (which is OK). I hope to keep this thread of interest for newbie gardeners and advanced horticulturalists alike.

By lamps, I include incadescents, fluorescents, metal halide, high pressure sodium, LEDs, and any new fangled invention that may be out there. Solar and filtered solar is also included

By biology, I mean leggy seedlings, premature bolting or internode length; as mediated by the red sensor phytochrome and the blue sensor cryptochrome. Since I grow leafy veggies, I consider bolting as a "negative", so I want to delay flowering. Others on this list want to speed up flowering and increase its extent; discussion of light color from this "positive" perspective is also welcome in this thread.

I would like to exclude the effect of color spectrum on photosynthesis per se, and leave that in the old thread "Why not blue?". I probably want to exclude photoperiod as well; this is covered in other threads.

To get everyone up to speed, let me repost a link first posted in "why not blue?", where solar light is filtered with dyes or colored plastic films.

Here is a link that might be useful: Clemson Colored Filters


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RE: Why not blue?

Oops, my first attempt to split this thread failed. I finally did get the above post into its own thread.

I plan to put UV issues into its own thread too.

Welcome back Zink! I have a question for you in the overdrive thread.


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RE: Why not blue?

> blue phosphors...don't last long...

This is a common belief, and there is some data to support it:
http://www.pets-warehouse.com/Aqualitchart.htm
...where a Philips Actinic 03 is compared before and after 3650 hours of use.
On the other hand, a "Hamiliton Super Actinic" seems to be holding up (but there was no "before" comparison given)

While comparing the spectra of reptile bulbs, I came across the following link:
http://www.saunalahti.fi/toweri/Pages/Spectrums.html

There is a comparison with the "Zoo Med ReptiSun 5.0 UVB" before and after 10 months use. After 10 months, the degradation in performance of the UVB, UVA, blue, green, orange, and red bands was relatively equal across the board. The degradation of any band was relatively small, perhaps ~20% of the original. The relative degradation of the spikey mercury emission peaks (~375, ~420 & ~440) seemed to be equivalent to the degradation of any phosphor. So most of this degradation may be due to loss of mercury vapor (absorbed by phosphor?), or aging of other lamp components such as the electrodes, rather than significant degradation of any of the phosphors per se. This lamp seems to have some halophosphor, supplemented with deluxe phosphors (~500nm & ~650nm), and of course some UV phosphors.

The generalized view that blue phosphors are the "bad boys" of the phosphor world may be too simplistic. If there are about 8-11 commercial blue phosphors, could 1 or 2 of these have given the rest an undeserved bad reputation?

One mechanism by which phosphors degrade is that excited ions of mercury are highly reactive and break down phosphors. If the phosphor is coated with a UV (254nm) transparent film, they could be resistant to chemical degradation.

I assume that this is what "Starcoat" technology is:
http://www.gelighting.com/na/business_lighting/education_resources/literature_library/sell_sheets/downloads/fluorescent/70714_starcoat_sxl.pdf
http://www.gelighting.com/na/business_lighting/education_resources/literature_library/sell_sheets/downloads/fluorescent/28345_starcoat_t5_linear.pdf

I need to do some more thinking along these lines.


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