Go back and re-read my post.
Mu does depend completely on the cloud position. But for the reasons I explain, the position does not change much with heater voltage.
In pure metal cathodes, the cathode work function (which is presumably what you mean by "barrier potential" - it isn't a potential, it's an energy) is a constant specific to the metal and independent of temperature. In oxide coated cathodes it is very weakly dependent on temperature.
Thermionic emision is much like evaporation of a liquid phase into a gas phase. For evaporation there is a "barier" to be overcome - termed the heat (energy) of evaporation. In thermionic emission a "work function" has to be overcome. In SI the relavent unit is the joule. For oxide coated cathodes it is around 2.4 x 10^-19 J. However in certain calculations it is more convenient to use the equivant energy unit electron-volt.
Far from suggesting there is no cloud, the fact that mu is MUCH higher than the tube physical grid-cathode and anode cathode ratio suggests is strong evidence that the cloud a) exists, and b) its' mean distance is some distance out from the cathode.
Carefull. In the 1930's manufacturers discovered that a small amount of silicon added to the cathode or filament metal improved emission. Trouble is, when zero or very low anode currents are drawn, an interface layer forms between the base metal and the oxide coating as the silicon difuses toward the metal surface. The when you attempt to draw anode current the tube behaves as though a large resistance is in series with the cathode, biasing the tube some way or even almost all teh way to cut-off.
Tubes made after WW2 up until the mid 1950's are likely to suffer from this effect. (except for thoriated tungsten tubes of course).
I am quite aware of the hazard of Interface Layer Formation (Zwischenschichtbildung for informative German references).
The cut-off bias is only to be used during the few seconds warm-up of a DHT.
The kT comes from the space-charge smoothing factor, which when inserted into Schottky's theorem converts the shot noise into Johnson form, so it can be thought of as "the Johnson noise of the cathode impedance". A little contrived I suppose, but that's convention for you. Shot noise power density is proportional to gm, which is in turn approximately proportional to the square root of the cathode current.
Actually I suspect that the 'shot noise' seen in normal valve operation is really Johnson noise (from the temperature of the space charge, which will not necessarily be the same as the cathode surface temperature), as the space charge smoothing calculation seems to remove all traces of eIa and magically insert kT in its place. Only under temperature-limited conditions do you see true shot noise. Just my theory!
You can read Schottky's original paper. You can read a multitude of books on the subject. As I said before, you can also read books on the measurement of noise factor, or the manuals for noise factor meters. Instrument manuals won't give you mathematical derivations, but they will tell you that shot noise when a space charge/electron cloud is present is reduced by the space charge.
The derivation of the formula for shot noise is treated extensively by A J Rack in BSTJ Vol 17:4, 1938, pages 592 to 619. Rack takes a completely different approach to Schottky and ends up with the same result. I quote from Rack's opening paragraph:
Several papers have shown that shot noise is decreased by the space charge.
The approach taken by Schottky was to prove an equation valid for both space charge limitted and temperature limitted conditions. This equation is:-
in = (2 q F^2 Ia df)^0.5
where in = RMS noise current, q is electron charge, Ia is the anode current, df is the measurement bandwidth.
For temperature limitted operation (ie anode takes all the emission and no electron cloud build up occurs), F = 1.
For space charge operation (ie anode does not take all the emission, and an electron cloud is established), F is defined by an equation dependent on the ratio of actual anode current to the temperatue limitted current, the cathode temperature, and the anode voltage, and is less than 1. There is a kT divisor term. As T (cathode temperature) increases, and as the anode current becomes small compared to the temperature limitted value, the kT term is taken out of the picture.
I suspect though, that where you saw a kT term in any discussion about noise, they were talking about thermal noise arising in circuit resistances (including the cathode radial resistance), and not shot noise.
Another way you may have been confused or read things wrong is the common pratice in engineering texts to relate all forms of white noise (shot, partition, thermal) to an equivalent "noise resistance" imagined to be in series with the grid of the tube in question, the tube considered to be perfectly noise free apart form the imagined series resistance. As shot noise, partition noise, and thermal (johnson) noise are all white noise ie have a gausian probability density function, one can then use the standard noise formula relating temperature to the noise power in a resistance and calculate an "equivalent noise temperature" of the tube. This does involve a kT term. In no way does it imply that shot noise depends on kT.
I remind you that if you wish to assert that the concept of a dense electron cloud, a concept well established for about the last hundred years, you must come up with a coherent rational and detailed explanation for ALL the things that the concept explains, which include:-
1) triode mu much greater than indicated by tube element spacings
2) shot noise not following root anode current below saturation
3) retarding field current having a slope according to log anode voltage
(the density of the electron cloud decreases exponentially with distance. Making the anode more negative pushes the electron cloud back toward and into the cathode, exposing the anode to a lower electron density)
4) retarding field current increases with decreasing anode-cathode distance, up to a maximum
5) there is a grid-cathode impedance comprising a capacitance and resistance in series, this RC impedance in parallel with the static grid-cathode capacitance. (explained in my post 76)
It's no good just stating there is no cloud, or stating that shot noise is thermal noise arising in the space charge (What? So you think there is a space charge then? What on earth do you think a space charge is? It's a cloud of electrons of course). You need to explain ALL items 1 thru 5 above. Not just one. ALL of them. Including how the various things vary with applied voltages and tube dimensions.
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The space charge smoothing factor is generally accepted to be:
F2 = 3k(0.644Tk)/(qVa)
For a perfectly concentric triode this becomes:
F2 = 2k(0.644Tk)gm/(qIa)
Which when subbed into Schottky leaves:
in = (4k(0.644Tk)gm df)0.5
I think this is probably where luckythedog has seen the kT term. Since real triodes are not geometrically ideal, there is also a fudge factor normally added:
in = (4k(0.644Tk)(gm/s) df)0.5
Where s (actually sigma, but I don't know how do a sigma symbol!) is typically found to be between 0.6 and 1.
Shot noise power therefore does vary with the square root of anode current, inasmuch as gm varies with the square root of anode current.
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That looks like some variant of A J Rack's approach in the BSTJ. The 0.644 constant is Racks' constant.
Schottky, (and Ballantine as I recall) as I said, took a diffferent approach, and came up with the more immediately useful:
F = 1.39 / [Ln(Is/Ip) - eVp/kT]^0.5.
Although it looks completely different to Rack's equation, when you evaluate it numerically, you get the same result for practical cases, as Rack said.
It is seen that as the anode current Ip goes down (Is is the temperature limitted ie saturation current), F goes toward zero.
I didn't include this equation in my last post for two reasons.
1) I hate doing equations in this forum, you can't do commonly used symbols as you said, and they end up hard to read - other forums have a decent way of displaying equations properly.
2) Despite Lucky saying how good Herrmann/Wagener is, I don't think he understands a word of it. Otherwise he would not have claimed it supports his wacky ideas, and he would have posted the calculation he claims to have done. If he doesn't understand H/W, he's not going to understand Schottky or Rack, and probably not much math in any case.
Some people skim read advanced texts and get some sort of feeling about what the author is on about without really understanding it. I do it myself before reading again properly line by line. A skim can mislead you.
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If shot noise is expressed in kT terms (as equivalent cathode impedance) one has to be everso careful about what temperature means. Thermionic emission is endothermic and the cathode by definition has lowered entropy, mean random energy and temperature. And because of the oft associated exponential term, small things matter- put it simply one would expect a colder than predicted cathode impedance to be a lower noise source, right............? Very tricky, very naughty and easy to trip with interpretation methinks.
Yup, temperature of free charge carriers seems the way to fly if one must. But that has its own trickiness. What's wrong with good old Gaussian shot noise ?
In = (2 q Ia BW)^0.5
........this must always apply, are we really suggesting it doesn't ? Surely not.
Yup, temperature of free charge carriers seems the way to fly if one must. But that has its own trickiness. What's wrong with good old Gaussian shot noise ?
In = (2 q Ia BW)^0.5
........this must always apply, are we really suggesting it doesn't ? Surely not.
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Yes, that should work with the 'right' T, one could also work backwards and find kT then one would know for sure. There is a certain catch 22 about it all IMO.
I have already dealt with that irrelevance in previous posts. A lower cathode temperature, however it has been lowered, merely reduces the electron cloud density. If the anode current is never-the-less lower than the emission capability at that lowered temperature, the shot noise amplitude still follows root anode current. And in practical cases of oxide coated cathodes, the radial resistance introduces introduces I^2.R dissipation that tends to cancell out the thermionic emission cooling. In other words, the noise amplitude cf root Ia does not drop with cathode temperature, it drops when Ia goes below Is.
Yes. Because it isn't In = (2 q Ia BW)^0.5; it's been proven to be
In = (2 q F^2 Ia BW)^0.5, where F=1 when anode current is temperature limitted, and F < 1 when it isn't. F tends toward zero as anode current (below the temperature limitted value) tends toward zero.
You asked for a reference. I gave you one.
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Maybe this is a good place to start. I suspect there might be various interpretations banded about. Space charge is a physics concept describing charge carriers in bulk distributed across a physical volume or space. Like a 3D version of charge distribution on surface of a sphere. For example, Ia might be described as physically conveyed by 'space charge' carriers. Any collection of charge carriers whose charge and volume is known comprises 'space charge'. It includes, but isn't limited to, thermionic electron emission into vacuum beyond its surface. [/QUOTE]
If one is modelling shot noise as an equivalent cathode impedance source it lowers that too. Even if the cathode is 'virtual'.
Where F is for 'fudge' 😉 But seriously, there's surely a fundamental here that gaussian statistics must apply. If there is an adjustment it must be because of interaction between discrete charge carriers, presumably as part of Ia in this case, hence F is small for low carrier concentrations (small Ia). Not necessarily anywhere near the cathode or related to emission............
Pretty sure no one will reply to this one … but isn't the problem really just a statistical one?
Let's say that turning ON a piece of equipment, twice a day (before work, after work) causes the thing to have a 10% shorter lifetime of its most sensitive and problem-prone tube, in 5 years of service. Well, that's 365 × 2 × 5 = 3,650 power-ups. Using mathematics that'd make Sister Josephine proud, 10% ÷ 3650 = 0.00274% worth of degradation per switching.
Looked at from that vantage, I can see an advantage that those clever TV designers in the early 1970s added to their sets by having the heaters of the tubes always on! Gets rid of that 10% in 5 years, at least from ion-bombardment degradation. Further, allows the TV's to turn on "instantly" … which they did. Maybe just having heater heated up … has its own tube-degradation, as gas molecules trapped in the heaters gradually migrate out (solid state diffusion!), thus again degrading the tubes over time. 10% in 5 years? More? Less? Who cares?
And of course the real obvious-man's solution is, "leave the equipment turned off". (LOL) You know, pretty furniture that impresses people when they come over, which can be turned on a few times a year to shake their pessimism, keep the power bill down, glow warmly next to the fireplace and so forth.
No, wait. We're supposed to listen to the marvels of our imagination, not just look at 'em.
GoatGuy
Let's say that turning ON a piece of equipment, twice a day (before work, after work) causes the thing to have a 10% shorter lifetime of its most sensitive and problem-prone tube, in 5 years of service. Well, that's 365 × 2 × 5 = 3,650 power-ups. Using mathematics that'd make Sister Josephine proud, 10% ÷ 3650 = 0.00274% worth of degradation per switching.
Looked at from that vantage, I can see an advantage that those clever TV designers in the early 1970s added to their sets by having the heaters of the tubes always on! Gets rid of that 10% in 5 years, at least from ion-bombardment degradation. Further, allows the TV's to turn on "instantly" … which they did. Maybe just having heater heated up … has its own tube-degradation, as gas molecules trapped in the heaters gradually migrate out (solid state diffusion!), thus again degrading the tubes over time. 10% in 5 years? More? Less? Who cares?
And of course the real obvious-man's solution is, "leave the equipment turned off". (LOL) You know, pretty furniture that impresses people when they come over, which can be turned on a few times a year to shake their pessimism, keep the power bill down, glow warmly next to the fireplace and so forth.
No, wait. We're supposed to listen to the marvels of our imagination, not just look at 'em.
GoatGuy
They do. So?
Yes. The ones in the cloud.
No. Not for low carrier concentrations. For high carrier concentrations, in the cloud (compared to the concentration after the grid or near the anode). The high concentration in the electron cloud comes about because the anode isn't drawing them all off. Duh!
Just to lighten the mood, I'll add that 1950s Juke Boxes sometimes started up with double voltage on the heaters. Once cathode current established in the power tubes a relay flipped over to normal heater voltage.
The idea was to get 'fast warm-up' so that you didn't have to wait for the music after you put in your dime.
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In your terms there is a physical cloud of charge which by convention alters 'virtual' potential and location of the cathode, and brings about adjustment to mu. I'm exploring whether there might be effectively no physical 'cloud' in a grid valve under negative bias, or in a diode for that matter - rather cathode potential adjustment happens at the cathode surface and altered initial velocity is a surrogate for 'virtual cathode' location. Same effect different mechanisms. See Herrmann/Wagener 1951 how both work function and external field together define emission (exponentially), potential barrier and intial velocity.
When deciding which, if either, of these mechanisms applies in this case, seems more problems with a physical cloud. Not least, mu should depend on electron cloud, as you say. Then emission should readily alter mu, and we all know how massively sensitive emission is to cathode temperature - it should be easy to bring about variation in mu by varying cathode temperature by small amounts. But it isn't.
But the elephant in the room I think here is the massive potential that would be represented by any such 'dense cloud' of charge carriers between cathode and grid. Working back from the 2.5x rule again to find the supposed density suggests such a cloud would be obviously disruptive. I can't find any reference to such a cloud in literature in a grid valve in normal operation, but if you know its alleged potential, density or location bring it on, Keit !
Nope, it more likely falls because carrier concentration in Ia falls with decreasing Ia. Duh ! 😉
Have you worked out the likely potential associated with this electron cloud, BTW.................... ?
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