From solution to mother equation

[haruspex]

View attachment 275538

Ok, so y is the log of the intensity?
As I asked, what is the power (n) in that fit, and what is the difference between the two runs?

 

[naviakam]

Ok, so y is the log of the intensity?
As I asked, what is the power (n) in that fit, and what is the difference between the two runs?

1. yes y is the number of ions.

2. n=4

3. the difference is the gas pressure in which the capacitor discharges:
at 6mbar the spectrum fits with exp
at 10mbar it's a power fit

 

[haruspex]

yes y is the number of ions.

The "power fit" graph you posted is log-linear, so is y in the power equation the number of ions (intensity?) or the log of the number?

And does this mean your exponential graph is also after taking logs, so you really have log(intensity) is negative exponential?!

n=4

Exactly?

the difference is the gas pressure in which the capacitor discharges:
at 6mbar the spectrum fits with exp
at 10mbar it's a power fit

So the full story is y=y(x,p), where p is the gas pressure.
Extremely artificially, you could write
$y=P(p)y_1(x)+(1-P(p))y_2(x)$, where P(6mbar)=0 and P(10mbar)=1.
I'm not seriously suggesting that would be valid, but you get the idea.

 

[naviakam]

The "power fit" graph you posted is log-linear, so is y in the power equation the number of ions (intensity?) or the log of the number? If you were to plot x^-4 on log-linear I would expect a straight line.

And does this mean your exponential graph is also after taking logs, so you really have log(intensity) is negative exponential?!

Exactly?

So the full story is y=y(x,p), where p is the gas pressure.
Extremely artificially, you could write
$y=P(p)y_1(x)+(1-P(p))y_2(x)$, where P(6mbar)=0 and P(10mbar)=1.
I'm not seriously suggesting that would be valid, but you get the idea.

1. it's been plotted in log format, y is number only not log

2. 3.8<n<4.1

3. then what is the estimated potential leading to each term in the equation $y=P(p)y_1(x)+(1-P(p))y_2(x)$?

 

[haruspex]

it's been plotted in log format, y is number only not log

Why would you plot it log-linear if trying to demonstrate a power law fit? Plot y-y₀ log-log against x, or plot y-y₀ against $x^{-4}$.

3. then what is the estimated potential leading to each term in the equation $y=P(p)y_1(x)+(1-P(p))y_2(x)$?

As I wrote, I am not seriously suggesting that is the correct form. I was just showing you how, given a function of two variables, the dependence on one might look very different for a different value of the other. To get any reliable sense of what is going on you would need results for several intermediate values of the pressure.

Anyway, what is this potential you are asking about? The voltage on the capacitor? Some potential function?

 

[naviakam]

Why would you plot it log-linear if trying to demonstrate a power law fit? Plot y-y₀ log-log against x, or plot y-y₀ against $x^{-4}$.

As I wrote, I am not seriously suggesting that is the correct form. I was just showing you how, given a function of two variables, the dependence on one might look very different for a different value of the other. To get any reliable sense of what is going on you would need results for several intermediate values of the pressure.

Anyway, what is this potential you are asking about? The voltage on the capacitor? Some potential function?

I was wondering if from the spectra we can guess a general potential function for each of the following equations:

 

[haruspex]

I was wondering if from the spectra we can guess a general potential function for each of the following equations:
View attachment 275586
View attachment 275587

As I indicated, I don't know what you mean by a "potential function" in this context.
Perhaps you could illustrate by starting from the other end: pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to.

Also, please post your "power fit" data as a graph of ln(y-y₀) against ln(x).

 

[naviakam]

As I indicated, I don't know what you mean by a "potential function" in this context.
Perhaps you could illustrate by starting from the other end: pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to.

Also, please post your "power fit" data as a graph of ln(y-y₀) against ln(x).

1. the capacitor discharge voltage is 10 kV, but ions are accelerated to MeV by an unknown potential. Then we try to guess this accelerating potential (starting from the other side: from spectra to potential!).

2. " pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to. "
Would you please help me how to do this.

3. here is the raw data:

1026.35 2.12E+10
973.90 2.28E+10
925.86 2.53E+10
881.75 2.73E+10
841.16 3.43E+10
803.73 4.18E+10
769.15 5.28E+10
737.15 6.57E+10
707.47 8.61E+10
679.92 1.15E+11
654.29 1.42E+11
630.42 1.89E+11
608.16 2.57E+11
587.38 3.53E+11
567.95 4.66E+11
549.77 6.18E+11
532.74 8.32E+11
516.77 1.10E+12
501.78 1.52E+12
487.70 1.90E+12
474.47 2.60E+12
462.02 3.33E+12

 

[haruspex]

1. the capacitor discharge voltage is 10 kV, but ions are accelerated to MeV by an unknown potential. Then we try to guess this accelerating potential (starting from the other side: from spectra to potential!).

2. " pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to. "
Would you please help me how to do this.

3. here is the raw data:

1026.35 2.12E+10
973.90 2.28E+10
925.86 2.53E+10
881.75 2.73E+10
841.16 3.43E+10
803.73 4.18E+10
769.15 5.28E+10
737.15 6.57E+10
707.47 8.61E+10
679.92 1.15E+11
654.29 1.42E+11
630.42 1.89E+11
608.16 2.57E+11
587.38 3.53E+11
567.95 4.66E+11
549.77 6.18E+11
532.74 8.32E+11
516.77 1.10E+12
501.78 1.52E+12
487.70 1.90E+12
474.47 2.60E+12
462.02 3.33E+12

Are these the data that you claim a power law for?
For those data I get a decent straight line fit between ln(I) and V. Just a bit of a curve at one end. Looks like I is proportional to exp(-0.02V).

If it is not the "power law" data, please post that too.

 

[naviakam]

Are these the data that you claim a power law for?
For those data I get a decent straight line fit between ln(I) and V. Just a bit of a curve at one end. Looks like I is proportional to exp(-0.02V).

If it is not the "power law" data, please post that too.

This data and many others fit well with power law without any curve at the end.

However what I want to know is that if we have power law and exponential equations (as stated in post #39) how we can assign a potential function to them? This is what we need to present to the class! Please help on this.

Thanks

 

[haruspex]

This data and many others fit well with power law without any curve at the end.

Do you mean a power law or a polynomial?

how we can assign a potential function to them?

As I keep telling you, I don't even understand that question. Presumably you do have some idea of how a potential function would lead to an intensity versus voltage function, so give me an example. Pick some arbitrary potential function and show what I=f(V) it would lead to.

 

[naviakam]

Do you mean a power law or a polynomial?

As I keep telling you, I don't even understand that question. Presumably you do have some idea of how a potential function would lead to an intensity versus voltage function, so give me an example. Pick some arbitrary potential function and show what I=f(V) it would lead to.

If we have $Y=KX^{-n}$ where Y is number of particles and X is their energy.
Then we have a potential like $V=CI^{4}$ where V is the potential, C is a constant and I is the current.
The potential lead to the particles to be accelerated!

How this two equations could be related mathematically?

 

[haruspex]

If we have $Y=KX^{-n}$ where Y is number of particles and X is their energy.
Then we have a potential like $V=CI^{4}$ where V is the potential, C is a constant and I is the current.
The potential lead to the particles to be accelerated!

How this two equations could be related mathematically?

What current is this? Is it the current carried by the flow of ions?
If so, seems to me that you are not looking for a "potential function" but rather two functions of potential: how the current (rate of ions) and the energy per ion depend on an applied potential.
I.e. I=I(V) and E=E(V) lead to an observed relationship between I and E, and you want to find out what I(V) and E(V) are.
Am I close?

 

[naviakam]

What current is this? Is it the current carried by the flow of ions?
If so, seems to me that you are not looking for a "potential function" but rather two functions of potential: how the current (rate of ions) and the energy per ion depend on an applied potential.
I.e. I=I(V) and E=E(V) lead to an observed relationship between I and E, and you want to find out what I(V) and E(V) are.
Am I close?

Yes you are right, then how it is formulated mathematically?

 

[haruspex]

Yes you are right, then how it is formulated mathematically?

There is clearly no unambiguous way to extract two relationships from the one, but an obvious guess is that in a vacuum the energy would be directly proportional to the voltage, leaving you with a simple way to extract the relationship between voltage and current from the observed data.
But you got different results at some other pressure, so this is not a vacuum. Presumably the gas present saps some of the KE, but should not affect the current.

 

[naviakam]

There is clearly no unambiguous way to extract two relationships from the one, but an obvious guess is that in a vacuum the energy would be directly proportional to the voltage, leaving you with a simple way to extract the relationship between voltage and current from the observed data.
But you got different results at some other pressure, so this is not a vacuum. Presumably the gas present saps some of the KE, but should not affect the current.

Not sure how to calculate I(V) and E(V).

$E=ItV, V=CI^{4}, Q=KC^{-n}, Y=KE^{-n}$
Then
$Y=Q(I^5tV)^{-n}$
resulted in
$V=(QY)^{-n}(I^5t)^{-1/n}$
where t is the time, E energy, Y intensity, V potential, Q constant, I current.
If correct how it is written more in the math style?
What if the equation in post #39 is considered, how rewritten considering such potential?

 

[haruspex]

$E=ItV,$

Isn't E here the energy of an individual ion?

But I am worried about my earlier interpretation. Your data say the current (ions per unit time) and energy (per ion) are negatively correlated. I cannot imagine what physical set up would lead to that. An applied potential that increases one should increase the other, no?

Is it possible for you to provide a far more detailed description of the apparatus? What is being varied to get the different readings?

 

[haruspex]

This data and many others fit well with power law without any curve at the end.

I assume the two columns in the data you posted are respectively x, y.
I just tried plotting y against x^-4. Nothing like a straight line.
Please post exactly what power law equation (constants included) you get for the relationship.

 

[naviakam]

Isn't E here the energy of an individual ion?

But I am worried about my earlier interpretation. Your data say the current (ions per unit time) and energy (per ion) are negatively correlated. I cannot imagine what physical set up would lead to that. An applied potential that increases one should increase the other, no?

Is it possible for you to provide a far more detailed description of the apparatus? What is being varied to get the different readings?

It is a plasma device called plasma focus consisting a chamber and electrode assembly of anode at the center and cathode around it. capacitor discharge (10 KV) in the gas filled the chamber accelerates the ions to MeV with the spectrum I sent the data earlier. The intensity of low energy ions are high and decreases with energy by $E^{-n}$ where n is around 4.

 

[naviakam]

I assume the two columns in the data you posted are respectively x, y.
I just tried plotting y against x^-4. Nothing like a straight line.
Please post exactly what power law equation (constants included) you get for the relationship.

This is another plot with power law fit in origin for the same device:
$Y=KX^{-n}$ where $K=6.8*10^{23}$ and $n=4.56$

 

[haruspex]

This is another plot with power law fit in origin for the same device:
$Y=KX^{-n}$ where $K=6.8*10^{23}$ and $n=4.56$

View attachment 275997

I need the raw data and the power law equation you believe fits it for the same data.
The fit above doesn't look that great to me.

 

[naviakam]

I need the raw data and the power law equation you believe fits it for the same data.
The fit above doesn't look that great to me.

power law was mentioned above and the data is:
1106.16 1.20E+10
1048.02 1.19E+10
994.96 1.27E+10
946.43 1.48E+10
901.95 1.92E+10
861.07 2.17E+10
823.44 2.79E+10
788.72 3.35E+10
756.65 4.31E+10
726.96 5.83E+10
699.44 7.19E+10
673.89 8.53E+10
650.14 9.91E+10
628.04 1.17E+11
607.45 1.32E+11
588.24 1.54E+11
570.31 1.79E+11
553.55 2.07E+11
537.87 2.30E+11
523.20 2.74E+11
509.46 3.13E+11
496.58 3.45E+11
484.51 3.91E+11
473.18 4.23E+11
462.54 4.83E+11
452.56 4.94E+11

 

[haruspex]

power law was mentioned above and the data is:
1106.16 1.20E+10
1048.02 1.19E+10
994.96 1.27E+10
946.43 1.48E+10
901.95 1.92E+10
861.07 2.17E+10
823.44 2.79E+10
788.72 3.35E+10
756.65 4.31E+10
726.96 5.83E+10
699.44 7.19E+10
673.89 8.53E+10
650.14 9.91E+10
628.04 1.17E+11
607.45 1.32E+11
588.24 1.54E+11
570.31 1.79E+11
553.55 2.07E+11
537.87 2.30E+11
523.20 2.74E+11
509.46 3.13E+11
496.58 3.45E+11
484.51 3.91E+11
473.18 4.23E+11
462.54 4.83E+11
452.56 4.94E+11

Is this the data that goes with the equation in post #55?

 

[naviakam]

Is this the data that goes with the equation in post #55?

Yes.

 

[naviakam]

$Y=P(p)Y1(E)+(1−P(p))Y2(E)$, where P(6mbar)=0 and P(10mbar)=1.
How to calculate Y(V) and E(V)?
Potential is $V=YE/It$.
where t is the time, E energy, Y intensity, V potential, I current.

 

[haruspex]

Yes.

One thing this thread has shown me is how hard it can be to discriminate between an exponential and a power law just based on the data.
I went back to your data in post #43 and found very good fits for both types. So I plotted the one against the other over the range of the x values, 460 to 1026. I.e. y_exp against y_pow where:
$y_{exp}=e^{-\frac x{50}}$ and $y_{pow}=x^{-10}$.
The graph is an excellent fit to $y_{pow}=2.28*10^{-23}y_{exp}+1.01*10^{-29}$. I found this very hard to believe, but I cannot see any errors in my work.
Of course, it could not remain so straight as x approaches zero since y_pow would shoot off to infinity.

I strongly suspect a general principle here, but I haven't figured out the details. Probably something to do with which terms in the expansion of e^-λx dominate for a given x.From this I suggest that you should treat all sets of data as being of the same type, either power or exp, and see which fits better overall.
I have to agree that for the data in post #57 power law looks more persuasive than exponential, so maybe take all as being power.Edit:
Tried other powers, an x range of 500 to 1000, and observed a general behaviour that $x^{-n}$ v. $e^{-knx}$ is a pretty straight line where k=0.00128 and n ranges from 4 to 12. But if I change the range of x (still over an octave) it breaks down quite fast; probably need k to be a function of a, where the x range is a to 2a.

 

[naviakam]

One thing this thread has shown me is how hard it can be to discriminate between an exponential and a power law just based on the data.
I went back to your data in post #43 and found very good fits for both types. So I plotted the one against the other over the range of the x values, 460 to 1026. I.e. y_exp against y_pow where:
$y_{exp}=e^{-\frac x{50}}$ and $y_{pow}=x^{-10}$.
The graph is an excellent fit to $y_{pow}=2.28*10^{-23}y_{exp}+1.01*10^{-29}$. I found this very hard to believe, but I cannot see any errors in my work.
Of course, it could not remain so straight as x approaches zero since y_pow would shoot off to infinity.
View attachment 276217
I strongly suspect a general principle here, but I haven't figured out the details. Probably something to do with which terms in the expansion of e^-λx dominate for a given x.From this I suggest that you should treat all sets of data as being of the same type, either power or exp, and see which fits better overall.
I have to agree that for the data in post #57 power law looks more persuasive than exponential, so maybe take all as being power.Edit:
Tried other powers, an x range of 500 to 1000, and observed a general behaviour that $x^{-n}$ v. $e^{-knx}$ is a pretty straight line where k=0.00128 and n ranges from 4 to 12. But if I change the range of x (still over an octave) it breaks down quite fast; probably need k to be a function of a, where the x range is a to 2a.

There are many other data sets best fitted with either power law or exponential, therefore $Y=P(p)Y1(E)+(1−P(p))Y2(E)$ still holds.
Considering the potential in post #60, how mathematically Y(V) and E(V) are formulated?

 

[haruspex]

There are many other data sets that are best fitted with either power law or exponential, therefore $Y=P(p)Y1(E)+(1−P(p))Y2(E)$ still holds.
Considering the potential in post #60, how mathematically Y(V) and E(V) are formulated?

Please post a dataset which clearly is closer to exponential than to a power law.

$Y=P(p)Y1(E)+(1−P(p))Y2(E)$ doesn't work because you have various powers and probably various coefficients in the exponentials.

 

[naviakam]

Please post a dataset which clearly is closer to exponential than to a power law.

$Y=P(p)Y1(E)+(1−P(p))Y2(E)$ doesn't work because you have various powers and probably various coefficients in the exponentials.

 

[naviakam]

Please post a dataset which clearly is closer to exponential than to a power law.

$Y=P(p)Y1(E)+(1−P(p))Y2(E)$ doesn't work because you have various powers and probably various coefficients in the exponentials.

This is another data fits well with Exp:
1205.36 1.69E+10
1135.78 1.72E+10
1072.51 1.81E+10
1014.83 1.79E+10
962.07 1.81E+10
913.71 1.79E+10
869.25 1.91E+10
828.31 2.04E+10
790.51 2.09E+10
755.54 2.32E+10
723.14 2.73E+10
693.06 2.95E+10
665.08 3.73E+10
639.02 4.24E+10
614.71 5.28E+10
592.01 6.60E+10
570.77 8.40E+10
550.88 1.11E+11
532.22 1.47E+11
514.71 1.86E+11
498.25 2.40E+11
482.77 3.15E+11
468.19 4.00E+11
454.45 5.07E+11

 

[haruspex]

This is another data fits well with Exp:
1205.36 1.69E+10
1135.78 1.72E+10
1072.51 1.81E+10
1014.83 1.79E+10
962.07 1.81E+10
913.71 1.79E+10
869.25 1.91E+10
828.31 2.04E+10
790.51 2.09E+10
755.54 2.32E+10
723.14 2.73E+10
693.06 2.95E+10
665.08 3.73E+10
639.02 4.24E+10
614.71 5.28E+10
592.01 6.60E+10
570.77 8.40E+10
550.88 1.11E+11
532.22 1.47E+11
514.71 1.86E+11
498.25 2.40E+11
482.77 3.15E+11
468.19 4.00E+11
454.45 5.07E+11

Again, I get a tolerable fit with either exp or power:

I slightly prefer the power fit; it seems to have less of a twist near the origin.

One of the tricky parts about fitting curves to data is that the best fit depends on what you choose to plot against what. E.g. the best least squares fit for $y=Ae^{kx}$ will be different from the best for $\ln(y)=\ln(A)+kx$. The latter model will put more emphasis on the match for the smaller values of y.
To optimise this, it helps if you have some idea how the accuracy of the measurements varies across the range.

Now, this is not an area I know much about, but I took a look at the black body spectrum formula. I note that this mixes power and exponential, so maybe something similar is going on here.

 

[naviakam]

Again, I get a tolerable fit with either exp or power:
View attachment 276269
I slightly prefer the power fit; it seems to have less of a twist near the origin.

One of the tricky parts about fitting curves to data is that the best fit depends on what you choose to plot against what. E.g. the best least squares fit for $y=Ae^{kx}$ will be different from the best for $\ln(y)=\ln(A)+kx$. The latter model will put more emphasis on the match for the smaller values of y.
To optimise this, it helps if you have some idea how the accuracy of the measurements varies across the range.

Now, this is not an area I know much about, but I took a look at the black body spectrum formula. I note that this mixes power and exponential, so maybe something similar is going on here.

Yes black body spectrum consists of a power term multiplied by exponential. It is intensity against wavelength compared with our case which is intensity against energy. In black body the total power is proportional to $T^4$ where T is temperature, and our potential is proportional to $I^4$ where I is current.

Then, how black body spectrum formula and our spectrum could be related?

Or, back to my first question, how our spectrum could be connected to the potential mathematically?

 

[haruspex]

Then, how black body spectrum formula and our spectrum could be related?

Or, back to my first question, how our spectrum could be connected to the potential mathematically?

As I have indicated, I do not know enough about the physics involved to answer questions like that.
Presumably you do have some such understanding, which is why I asked you to show me an example of how some given potential would lead to a particular spectrum. If you cannot do that we're both in the dark.

What I can do is analyse your data and suggest ways to unify the results into a consistent relationship.
How many data sets do you have? If you care to post more I will study them.

 

[naviakam]

As I have indicated, I do not know enough about the physics involved to answer questions like that.
Presumably you do have some such understanding, which is why I asked you to show me an example of how some given potential would lead to a particular spectrum. If you cannot do that we're both in the dark.

What I can do is analyse your data and suggest ways to unify the results into a consistent relationship.
How many data sets do you have? If you care to post more I will study them.

As requested in post #62:
considering intensity $Y=P(p)Y1(E)+(1−P(p))Y2(E)$ (where $Y1$ and $Y2$ are the power and exp functions respectively) and potential $V=YE/It$, how mathematically $Y(V)$ and $E(V)$ are formulated/written?