User:Alexander T. J. Barron/Notebook/PHYC 307L Lab Notebook/planck notes

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Contents

Planck's Constant Lab

Setup

PASCO Scientific Hg Light Source, model OS-9286

Wavetek True RMS multimeter, model 85XT

TEKTRONIX TDS 1002 Two Channel Digital Oscilloscope

Test batteries.... check.


Data

Experiment 1A

Green Line

100% -

1) 0.836 V - t = 10.19 s

2) 0.836 V - t = 10.61 s

80% -

1) 0.837 V - t = 17.25 s

2) 0.837 V - t = 12.85 s

60% -

1) 0.835 V - t = 13.48 s

2) 0.836 V - t = 17.95 s

40% -

1) 0.836 V - t = 24.45 s

2) 0.836 V - t = 21.23 s

20% -

1) 0.834 V - t = 33.74 s

2) 0.835 V - t = 43.80 s


Note that our voltage readings vary over the course of data taking. This is probably not desirable, since the intensity shouldn't alter the canceling potential. We might have to come back and retake this data.

Yellow Line

100% -

1) 0.704 V - t = 7.82 s

2) 0.705 V - t = 13.34 s

3) 0.705 V - t = 11.31 s

80% -

1) 0.705 V - t = 10.12 s

2) 0.705 V - t = 10.54 s

3) 0.706 V - t = 14.95 s

60% -

1) 0.705 V - t = 23.54 s

2) 0.706 V - t = 19.62 s

3) 0.706 V - t = 11.80 s

40% -

1) 0.704 V - t = 33.04 s

2) 0.704 V - t = 29.13 s

3) 0.704 V - t = 38.07 s

20% -

1) 0.700 V - t = 39.18 s

2) 0.700 V - t = 44.70 s

3) 0.700 V - t = 46.38 s


Experiment 1B

DVM

yellow-

1) 0.714 V

2) 0.711 V

3) 0.706 V

green-

1) 0.849 V

2) 0.830 V

3) 0.846 V

right blue-

1) 1.488 V

2) 1.499 V

3) 1.477 V

mid blue-

1) 1.710 V

2) 1.710 V

3) 1.722 V

left blue-

1) 2.053 V

2) 2.030 V

3) 2.067 V


Experiment 2

Battery test... check.

1st Order DVM

yellow-

1) 0.711 V

2) 0.710 V

green-

1) 0.840 V

2) 0.841 V

right blue-

1) 1.479 V

2) 1.480 V

mid blue-

1) 1.667 V

2) 1.671 V

left blue-

1) 1.994 V

2) 2.002 V

2nd Order DVM

Refocused lens to obtain distinct 2nd order spectra. Thought it prudent... don't want frequency overlap. The spectra lines are much wider than before.

yellow-

1) 0.650 V

2) 0.655 V

3) 0.663 V

4) 0.684 V

green-

1) 1.204 V

2) 1.229 V PHOTONS GONE WILD!!!

3) 1.221 V

4) 1.226 V

right blue-

1) 1.427 V

2) 1.435 V

3) 1.451 V

4) 1.470 V

mid blue-

1) 1.610 V

2) 1.628 V

3) 1.668 V

4) 1.675 V

left blue-

1) 1.992 V

2) 1.988 V

3) 2.023 V

4) 2.024 V

We decided to do 3 runs due to inconsistencies in data, but more inconsistency came with the 3rd run.

It seems like the voltage reading will just keep climbing no matter how long we wait... maybe there is a problem with the apparatus.


Data Analysis

Experiment 1 - Preliminaries

This data is used to show the physical realities of Planck's constant and the photo-electric effect qualitatively. We take data on the stopping potential for a frequency, then see how much time the capacitor in the apparatus takes to charge to the stopping potential. Rinse and repeat for another frequency.


Vstop vs. Transmission Charge Time vs. Transmission


Error bars represent 68% confidence intervals.

As one can see, the charge time increases with decreased transmission. This doesn't make sense at all using the photon theory of light. If this experiment strictly obeyed the theory of photons, the stopping potential of the apparatus would be the same regardless of how intense the incident light was. Hence, the charge time would be the same for any transmission rate. If the experiment obeyed the classical theory, charge time should increase with decreased transmission - the supposed amplitude of the waves should have decreased, so the "charged" photoelectrons ejected should have been less energetic. The observed trend correlates with the classical theory of light.

The voltage reading trend to drop with decreased transmission is at odds with the photon theory as well, via the above argument. The most plausible explanation for the drop I can think of involves the current leak from the capacitor in the apparatus. The manual creates the analogy of the capacitor being a bathtub being filled with "different flow rates while the drain is partly open." If the classical waves can't charge enough electrons quickly enough, the current leak could overcome the charging under low transmission filters.


The next preliminary experiment compares stopping potentials for differing frequencies.

Comparison for all First-Order Spectra

Clearly, the frequency axis increases to the right, as does the stopping potential. This supports the photon theory of light, since the intensity should be fairly constant over the spectra, indicating that intensity doesn't alter the energy of excited photoelectrons.

Experiment 2: Planck's Constant

Here we measure the stopping potential for all spectra in both the first order and second order series.

Via this illustration of measured spectra from the lab manual:


From Einstein's famous relation:

 E = eV + \omega_0  \longrightarrow  eV = h\nu - \omega_0

I can find h via linear regression.

Error methods:

A=\frac{\sum \frac{x_i^2}{\sigma_i^2} \sum \frac{y_i}{\sigma_i^2} - \sum \frac{x_i}{\sigma_i^2} \sum \frac{x_i y_i}{\sigma_i^2}}{\Delta} \mbox{,}~~~~~~~~~~~~~~~~~ \sigma_A^2 = \frac{1}{\Delta} \sum \frac {x_i^2}{\sigma_i^2}
B=\frac{\sum \frac{1}{\sigma_i^2} \sum \frac{x_i y_i}{\sigma_i^2} - \sum \frac{x_i}{\sigma_i^2} \sum \frac{y_i}{\sigma_i^2}}{\Delta} \mbox{,}~~~~~~~~~~~~~~~~~ \sigma_B^2 = \frac{1}{\Delta} \sum \frac {1}{\sigma_i^2}
\Delta=\sum \frac{1}{\sigma_i^2} \sum \frac{x_i^2}{\sigma_i^2} - \left (\sum \frac{x_i}{\sigma_i^2} \right)^2


Numerical Representations

1st-Order Spectra:

  • h = (4.52 ± 0.0052)e-15 [eV s]
  • ω0 = 1.64 ± 0.0031 [eV]

2nd-Order Spectra

  • h = (3.53 ± .039)e-15 [eV s]
  • ω0 = 0.90 ± 0.024 [eV]
accepted h = 4.14e-15 [eV s][1]
accepted ω0 = 1.36 ± 0.08 [eV][2]


Graphic Representations

Code


%% Planck's Constant Lab

% Alexander Barron
% Junior Lab Fall 2008

close all, clear all;

%% Load Data

% Experiment 1A

    % Green Line

    gtdata100 = [10.19 10.61];
    gtdata80 = [17.25 12.85];
    gtdata60 = [13.48 17.95];
    gtdata40 = [24.45 21.23];
    gtdata20 = [33.74 43.80];

    gVdata100 = [.836 .836];
    gVdata80 = [.837 .837];
    gVdata60 = [.835 .836];
    gVdata40 = [.836 .836];
    gVdata20 = [.834 .835];

    % Yellow Line

    ytdata100 = [7.82 13.34 11.31];
    ytdata80 = [10.12 10.54 14.95];
    ytdata60 = [23.54 19.62 11.80];
    ytdata40 = [33.04 29.13 38.07];
    ytdata20 = [39.18 44.70 46.38];

    yVdata100 = [.704 .705 .705];
    yVdata80 = [.705 .705 .706];
    yVdata60 = [.705 .706 .706];
    yVdata40 = [.704 .704 .704];
    yVdata20 = [.700 .700 .700];
    
% Experiment 1B

    Vy = [.714 .711 .706];
    Vg = [.849 .830 .846];
    Vrb = [1.488 1.499 1.477];
    Vmb = [1.710 1.710 1.722];
    Vlb = [2.053 2.030 2.067];

% Experiment 2

    % 1st Order DVM

    DVMy1st = [.711 .710];
    DVMg1st = [.840 .841];
    DVMrb1st = [1.479 1.480];
    DVMmb1st = [1.667 1.671];
    DVMlb1st = [1.994 2.002];

    % 2nd Order DVM

    DVMy2nd = [.650 .655 .663 .684];
    DVMg2nd = [1.204 1.229 1.221 1.226];
    DVMrb2nd = [1.427 1.435 1.451 1.470];
    DVMmb2nd = [1.610 1.628 1.668 1.675];
    DVMlb2nd = [1.992 1.988 2.023 2.024];
    
    
%% Mean and Std Err

% Experiment 1A

    % Green Line

    mugtdata100 = mean(gtdata100);
    errgtdata100 = std(gtdata100)*1/sqrt(length(gtdata100)-1);

    mugVdata100 = mean(gVdata100);
    errgVdata100 = std(gVdata100)*1/sqrt(length(gVdata100)-1);

    mugtdata80 = mean(gtdata80);
    errgtdata80 = std(gtdata80)*1/sqrt(length(gtdata80)-1);

    mugVdata80 = mean(gVdata80);
    errgVdata80 = std(gVdata80)*1/sqrt(length(gVdata80)-1);

    mugtdata60 = mean(gtdata60);
    errgtdata60 = std(gtdata60)*1/sqrt(length(gtdata60)-1);

    mugVdata60 = mean(gVdata60);
    errgVdata60 = std(gVdata60)*1/sqrt(length(gVdata60)-1);

    mugtdata40 = mean(gtdata40);
    errgtdata40 = std(gtdata40)*1/sqrt(length(gtdata40)-1);

    mugVdata40 = mean(gVdata40);
    errgVdata40 = std(gVdata40)*1/sqrt(length(gVdata40)-1);

    mugtdata20 = mean(gtdata20);
    errgtdata20 = std(gtdata20)*1/sqrt(length(gtdata20)-1);

    mugVdata20 = mean(gVdata20);
    errgVdata20 = std(gVdata20)*1/sqrt(length(gVdata20)-1);

    clear gtdata100 gtdata80 gtdata60 gtdata40 gtdata20;
    clear gVdata100 gVdata80 gVdata60 gVdata40 gVdata20;
    
    % Yellow Line

    muytdata100 = mean(ytdata100);
    errytdata100 = std(ytdata100)*1/sqrt(length(ytdata100)-1);

    muyVdata100 = mean(yVdata100);
    erryVdata100 = std(yVdata100)*1/sqrt(length(yVdata100)-1);

    muytdata80 = mean(ytdata80);
    errytdata80 = std(ytdata80)*1/sqrt(length(ytdata80)-1);

    muyVdata80 = mean(yVdata80);
    erryVdata80 = std(yVdata80)*1/sqrt(length(yVdata80)-1);

    muytdata60 = mean(ytdata60);
    errytdata60 = std(ytdata60)*1/sqrt(length(ytdata60)-1);

    muyVdata60 = mean(yVdata60);
    erryVdata60 = std(yVdata60)*1/sqrt(length(yVdata60)-1);

    muytdata40 = mean(ytdata40);
    errytdata40 = std(ytdata40)*1/sqrt(length(ytdata40)-1);

    muyVdata40 = mean(yVdata40);
    erryVdata40 = std(yVdata40)*1/sqrt(length(yVdata40)-1);

    muytdata20 = mean(ytdata20);
    errytdata20 = std(ytdata20)*1/sqrt(length(ytdata20)-1);

    muyVdata20 = mean(yVdata20);
    erryVdata20 = std(yVdata20)*1/sqrt(length(yVdata20)-1);
    
    clear ytdata100 ytdata80 ytdata60 ytdata40 ytdata20;
    clear yVdata100 yVdata80 yVdata60 yVdata40 yVdata20;
    
% Experiment 1B

    muVy = mean(Vy);
    errVy = std(Vy)*1/sqrt(length(Vy)-1);
    
    muVg = mean(Vg);
    errVg = std(Vg)*1/sqrt(length(Vg)-1);
    
    muVrb = mean(Vrb);
    errVrb = std(Vrb)*1/sqrt(length(Vrb)-1);
    
    muVmb = mean(Vmb);
    errVmb = std(Vmb)*1/sqrt(length(Vmb)-1);
    
    muVlb = mean(Vlb);
    errVlb = std(Vlb)*1/sqrt(length(Vlb)-1);
    
    clear Vy Vg Vrb Vmb Vlb;
    
    
% Experiment 2

    % 1st Order
    
    muDVMy1st = mean(DVMy1st);
    errDVMy1st = std(DVMy1st)*1/sqrt(length(DVMy1st)-1);
    
    muDVMg1st = mean(DVMg1st);
    errDVMg1st = std(DVMg1st)*1/sqrt(length(DVMg1st)-1);
    
    muDVMrb1st = mean(DVMrb1st);
    errDVMrb1st = std(DVMrb1st)*1/sqrt(length(DVMrb1st)-1);
    
    muDVMmb1st = mean(DVMmb1st);
    errDVMmb1st = std(DVMmb1st)*1/sqrt(length(DVMmb1st)-1);
    
    muDVMlb1st = mean(DVMlb1st);
    errDVMlb1st = std(DVMlb1st)*1/sqrt(length(DVMlb1st)-1);
    
    % 2nd Order
    
    muDVMy2nd = mean(DVMy2nd);
    errDVMy2nd = std(DVMy2nd)*1/sqrt(length(DVMy2nd)-1);
    
    muDVMg2nd = mean(DVMg2nd);
    errDVMg2nd = std(DVMg2nd)*1/sqrt(length(DVMg2nd)-1);
    
    muDVMrb2nd = mean(DVMrb2nd);
    errDVMrb2nd = std(DVMrb2nd)*1/sqrt(length(DVMrb2nd)-1);
    
    muDVMmb2nd = mean(DVMmb2nd);
    errDVMmb2nd = std(DVMmb2nd)*1/sqrt(length(DVMmb2nd)-1);
    
    muDVMlb2nd = mean(DVMlb2nd);
    errDVMlb2nd = std(DVMlb2nd)*1/sqrt(length(DVMlb2nd)-1);
    
    clear DVMy1st DVMg1st DVMrb1st DVMmb1st DVMlb1st DVMy2nd DVMg2nd...
         DVMrb2nd DVMmb2nd DVMlb2nd;
    

%% Preliminaries - Exp. 1

mugVvec = [mugVdata100 mugVdata80 mugVdata60 mugVdata40 mugVdata20];
errgVvec = [errgVdata100 errgVdata80 errgVdata60 errgVdata40 errgVdata20];

scrsz = get(0,'ScreenSize');
figure('Position',[1 scrsz(4)/1.5 scrsz(3)/1.55 scrsz(4)/1.6]);
subplot(1,2,1), errorbar(mugVvec,errgVvec,'ob');
title('Green Line');
ylabel('Stopping Potential [V]');
xlabel('% Transmission');

muyVvec = [muyVdata100 muyVdata80 muyVdata60 muyVdata40 muyVdata20];
erryVvec = [erryVdata100 erryVdata80 erryVdata60 erryVdata40 erryVdata20];

subplot(1,2,2), errorbar(muyVvec,erryVvec,'ob');
title('Yellow Line');
ylabel('Stopping Potential [V]');
xlabel('% Transmission');

mugtvec = [mugtdata100 mugtdata80 mugtdata60 mugtdata40 mugtdata20];
errgtvec = [errgtdata100 errgtdata80 errgtdata60 errgtdata40 errgtdata20];

figure('Position',[25 scrsz(4)/1.5 scrsz(3)/1.55 scrsz(4)/1.6]);
subplot(1,2,1), errorbar(mugtvec,errgtvec,'or');
title('Green Line');
ylabel('Charge Time [s]');
xlabel('% Transmission');

muytvec = [muytdata100 muytdata80 muytdata60 muytdata40 muytdata20];
errytvec = [errytdata100 errytdata80 errytdata60 errytdata40 errytdata20];

subplot(1,2,2), errorbar(muytvec,errytvec,'or');
title('Yellow Line');
ylabel('Charge Time [s]');
xlabel('% Transmission');

muallVvec = [muVy muVg muVrb muVmb muVlb];
errallVvec = [errVy errVg errVrb errVmb errVlb];

figure('Position',[50 scrsz(4)/1.5 scrsz(3)/1.55 scrsz(4)/1.6]);
errorbar(muallVvec,errallVvec,'ob');
title('Comparison of Spectra');
ylabel('Stopping Potential [V]');


%% Planck's Constant - Exp. 2

freqvec = [5.18672 5.48996 6.87858 7.40858 8.20264];
freqvec = freqvec.*10^14;

e = 1.60218e-19;

muDVM1stvec = [muDVMy1st muDVMg1st muDVMrb1st muDVMmb1st muDVMlb1st];
errDVM1stvec = [errDVMy1st errDVMg1st errDVMrb1st errDVMmb1st errDVMlb1st];

muDVM2ndvec = [muDVMy2nd muDVMg2nd muDVMrb2nd muDVMmb2nd muDVMlb2nd];
errDVM2ndvec = [errDVMy2nd errDVMg2nd errDVMrb2nd errDVMmb2nd errDVMlb2nd];



w1st = 1./errDVM1stvec.^2

Delta1st = sum(w1st)*sum(w1st.*freqvec.^2) - (sum(w1st.*freqvec))^2

A1st = (sum(w1st.*freqvec.^2)*sum(w1st.*muDVM1stvec)...
    - sum(w1st.*freqvec)*sum(w1st.*freqvec.*muDVM1stvec))/Delta1st

B1st = (sum(w1st)*sum(w1st.*freqvec.*muDVM1stvec)...
    - sum(w1st.*freqvec)*sum(w1st.*muDVM1stvec))/Delta1st

errA1st = sqrt((sum(w1st.*freqvec.^2))/Delta1st)

errB1st = sqrt(sum(w1st)/Delta1st)



w2nd = 1./errDVM2ndvec.^2

Delta2nd = sum(w2nd)*sum(w2nd.*freqvec.^2) - (sum(w2nd.*freqvec))^2

A2nd = (sum(w2nd.*freqvec.^2)*sum(w2nd.*muDVM2ndvec)...
    - sum(w2nd.*freqvec)*sum(w2nd.*freqvec.*muDVM2ndvec))/Delta2nd

B2nd = (sum(w2nd)*sum(w2nd.*freqvec.*muDVM2ndvec)...
    - sum(w2nd.*freqvec)*sum(w2nd.*muDVM2ndvec))/Delta2nd

errA2nd = sqrt((sum(w2nd.*freqvec.^2))/Delta2nd)

errB2nd = sqrt(sum(w2nd)/Delta2nd)



figure('Position',[75 scrsz(4)/1.5 scrsz(3)/1.55 scrsz(4)/1.6]);
subplot(1,2,1), errorbar(freqvec,muDVM1stvec,errDVM1stvec,'ok');
title('1st Order');
xlabel('Frequency [1/s]');
ylabel('Stopping Potential Energy [eV]');

hold on;

brdrx = .3e14;
brdry = .08;

xfit = [0 freqvec];
fit1st = A1st + B1st.*xfit;
plot(xfit,fit1st,'k');

fitup1st = A1st +(B1st+errB1st).*xfit;
plot(xfit,fitup1st,'b--');

fitdn1st = A1st + (B1st-errB1st).*xfit;
plot(xfit,fitdn1st,'b-.');

line([0 0],[A1st-errA1st A1st+errA1st],'Color','r');

legend('1st Order Data','Linear Fit','Upper Bound from best A',...
    'Lower Bound from best A','A error','Location','NorthWest');

line([-.55 .55].*10^13,[A1st+errA1st A1st+errA1st],'Color','r');
line([-.55 .55].*10^13,[A1st-errA1st A1st-errA1st],'Color','r');

xlim([-.55-brdrx, xfit(end)+brdrx]);
ylim([A1st-errA1st-brdry, fitup1st(end)+brdry]);

subplot(1,2,2), errorbar(freqvec,muDVM2ndvec,errDVM2ndvec,'ok');
title('2nd Order');
xlabel('Frequency [1/s]');
ylabel('Stopping Potential Energy [eV]');

hold on;

fit2nd = A2nd + B2nd.*xfit;
plot(xfit,fit2nd,'k');

fitup2nd = A2nd +(B2nd+errB2nd).*xfit;
plot(xfit,fitup2nd,'b--');

fitdn2nd = A2nd + (B2nd-errB2nd).*xfit;
plot(xfit,fitdn2nd,'b-.');

line([0 0],[A2nd-errA2nd A2nd+errA2nd],'Color','r');

legend('2nd Order Data','Linear Fit','Upper Bound from best A',...
    'Lower Bound from best A','A error','Location','NorthWest');

line([-.55 .55].*10^13,[A2nd+errA2nd A2nd+errA2nd],'Color','r');
line([-.55 .55].*10^13,[A2nd-errA2nd A2nd-errA2nd],'Color','r');

xlim([-.55-brdrx, xfit(end)+brdrx]);
ylim([A2nd-errA2nd-brdry, fitup2nd(end)+brdry]);

References

  1. Planck's Constant Wikipedia [Planck-Wikipedia]
  2. PASCO Tech Note 303. Link

    [PASCO-TechNote]

    Thanks to Paul Klimov for finding this on the company's website.

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