photoelectric effect

Photoelectric Effect Lab: Determination of Planck’s Constant and the Work Function of a Photocathode

Chrissy L. Porter and Erik J. Sánchez* Department of Physics, Portland State University, PO Box 751

Portland, OR 97207, USA (Dated: April 26, 2011)

A computer simulation was used to model the photoelectric effect. The program simulated monochromatic light of a chosen frequency striking a piece of metal and measured the current due to the resulting photo emission of electrons. By setting retarding voltages to inhibit the flow of the photocurrent, this program was used to find the stopping voltages for several wavelengths of light. These stopping voltages were plotted against the frequency of the incoming light to find the work function of the metal and to experimentally measure Planck’s constant. The value of Planck’s constant obtained was (6.743 ± .142) × 10−34 kg m/s, and the calculated work function for the photocathode used (#3 in the simulation) was (2.4193 ± .0737) × 10−19 J.

INTRODUCTION

First observed by Heinrich Hertz in 1887, the photo- electric effect proved to be an experimental demonstra- tion of the quantization of light. The observed effect is as follows: When monochromatic light with sufficiently high energy shines on a piece of metal, electrons are emit- ted from the metal with some amount of kinetic energy. If the frequency of the light is increased, the kinetic en- ergy of the liberated electrons increases; however, the kinetic energy is unchanged if the intensity of the inci- dent beam is increased. This phenomenon is due to the fact that electromagnetic energy is transferred in distinct units (quanta) according to the relation proposed by Max Planck in 1900,

E = hν, (1)

where the proportionality factor is Planck’s constant, h = 6.626 × 10−34 J ·s, E is the energy of a photon of light, and ν is the frequency of the photon.

Albert Einstein used Planck’s equation to explain the photoelectric effect in 1905. He stated that when an elec- tron in a piece of metal absorbs a photon with frequency ν, the photon gives up its energy (with quantity described by Eq. 2) to the electron. The electron then has enough energy to escape the metal. The kinetic energy of the escaped electron is

KE = hν − Φ, (2)

where Φ is the energy it takes the electron to just break away from the metal, called the work function. The pur- pose of this experiment is to use Eq. 2 to experimentally calculate Planck’s constant and compare it to the ac- cepted value, and also to calculate the work function of the simulated metal.

! ! ! !

!"#$%&"'()*+'+"(

&(,,(

-(

,,(

./(

FIG. 1: A schematic of the experimental setup emulated by the computer program used in this lab. Monochromatic light strikes a piece of metal, and electrons are emitted. The cur- rent decreases as Vr is increased until Vs is reached.

EXPERIMENTAL DESIGN AND THEORY

The experimental setup modeled by the author’s com- puter simulation1 is shown in Figure 1. A beam of monochromatic light strikes a metal. If the photons have an appropriate wavelength, electrons are emitted. A cur- rent is generated by electrons with enough kinetic energy crossing the gap to the other plate, completing the cir- cuit. The program includes six simulated photocathodes consisting of various metals with unique work functions. For this experiment, photocathode #3 was used.

To measure the kinetic energy of the escaping elec- trons, a retarding voltage, Vr, is applied. This causes the plate from which the electrons are emitted to be- come positively charged, making it more difficult for the electrons to cross the gap. When this retarding voltage is sufficiently strong, the flow of electrons ceases alto- gether. This point is called the stopping voltage, Vs, and it reflects the value of the kinetic energy of the liberated electrons, according to

2

KE = eVs, (3)

where e = 1.602176487 × 10−19 C is the charge of an electron. Substituting this expression for kinetic energy into Equation 2 and rearranging, one finds

Vs = h

e ν −

Φ

e . (4)

In this experiment, Equation 4 is employed to measure Planck’s constant and the work function of the metal by plotting several measured stopping voltages versus the corresponding frequencies of light. Using a weighted least squares regression, a fit line of the form yfit = A + B x is found for this data, and then Planck’s constant and the work function are determined via

h = eB (5)

and,

Φ = −eA. (6)

It should be noted that Equation 6 gives Φ in joules. The value of Φ in electron volts would simply be −A.

In this lab, all of the weighted least squares regressions used to find the parameters a and b in trend lines of the form yfit = a + bx were calculated “by hand” in Matlab using the following equations from Bevington and Robinson2:

∆ = ∑ 1

σ2i

∑ x2i σ2i

− (∑ xi

σ2i

)2 , (7)

a = 1

∆

(∑ x2i σ2i

∑ yi σ2i

− ∑ xi

σ2i

∑ xiyi σ2i

) , (8)

and

b = 1

∆

(∑ 1 σ2i

∑ xiyi σ2i

− ∑ xi

σ2i

∑ yi σ2i

) . (9)

Their variances were found using

σ2a = 1

∆

∑ x2i σ2i

, (10)

and

σ2b = 1

∆

∑ 1 σ2i

, (11)

from the same text.

PROCEDURE

Since this was a computer simulated experiment, the procedure consisted solely of the data acquisition and analysis protocols. The total current measured in the

simulation is the sum of background current and pho- tocurrent, and both of these currents have noise. Since photocurrent is what needed to be determined, it was first necessary to measure the average background cur- rent and the standard deviation, σback, in that cur- rent. This was done by acquiring current data using a wavelength that produces a very low stopping voltage, and Vr values significantly above that stopping voltage. The wavelength chosen was 7000 Å and voltages between 20 V and 39 V were used. Forty runs of twenty data points each were performed, for a total of 800 measure- ments of background current. These 800 measurements were then averaged and their standard deviation calcu- lated.

The main data collection procedure consisted of mea- suring twenty current values at increasing levels of the retarding voltage for nine different wavelengths. For each wavelength, 40 runs were performed using the same set of twenty Vr values each time. For each Vr value, the stan- dard deviation in the total current values, σtotal was cal- culated. After subtracting the average background cur- rent from all of the total current measurements, 40 pho- tocurrent measurements for each Vr level were acquired. These mean of these values was then found to acquire a set of 20 average photocurrent values at increasing re- tarding voltages for each of the nine wavelengths. The standard deviation in the average photocurrent measure- ments, σphoto, was calculated using the equation,

σphoto = √ σ2total + σ

2 back . (12)

The wavelengths were chosen to be distributed evenly within the available range of the simulated light source, and the Vr values were carefully chosen so that the full linearly decreasing region of the photocurrent would be captured (see Figure 3). The wavelengths and voltage steps chosen are listed in Table 1.

TABLE I: Voltage Steps for Chosen Wavelengths

Wavelength (Å) Voltage Start & Step (V)

2500 .2

3000 .15

3500 .125

4000 .09

4500 .075

5000 .0575

5500 .045

6000 .033

6500 .03

3

DATA AND DATA REDUCTION

The average background current, calculated in the manner described in the previous section, was found to be 1.6544×10−11 A. The standard deviation in this mea- surement, σback, was 35.7735 × 10−11 A. Having a stan- dard deviation larger than the mean is not ideal, but in this case is just a reflection of the fact that the back- ground current programmed into the simulation varied widely – often over one to two orders of magnitude.

For each of the nine wavelengths, a weighted least squares regression was performed on the average pho- tocurrent data points within the linearly decreasing re- gion, to find a line of the form y = a + bx. This allowed for the calculation of the stopping voltage, Vs, for each wavelength, where

Vs = −a b . (13)

Representative plots for the regression of the Iphoto vs. Vr data and determination of Vs are shown in Figures 2 and 3, for the two wavelengths representing the low and high end of the chosen spectrum. The error bars shown are based on the value for σphoto at each point.

0 0.5 1 1.5 2 2.5 3 3.5 4 0

1

2

3

4

5

6

7

8

R etar d i n g Vol tage (V )

P h

o t o c u

r r e n

t ( x

1 0

! 9

Å )

! = 2500 Å

V s t o p= 3. 5 2 7 2 V

FIG. 2: Photocurrent vs. Vr for 2500 Å light. Squares are the averaged values for Iphoto at each Vr. Error bars are given by that average ±σphoto. The line is a weighted least squares regression of points in the range where cur- rent decreases linearly with voltage. The fit line equation is: yfit = 8.7167 × 10−9 + (−2.4713 × 10−9) ν. The uncertainty in the y-intercept is σa = .9291 × 10−9 A and in the slope is σb = .3244 × 10−9 A/V .

The error bars in Figure 2 are of interest because they smoothly decrease in size with increasing retarding volt- age. This suggests that the computer program produces

some conical heteroskedacity, that is, that the random error programmed into the data actually decreases with increasing voltage. This phenomenon was observed in the data for every wavelength, but is most pronounced in this plot where the wavelength was at the limit of the photocathode’s emission range and thus the stan- dard deviation was a more considerable fraction of the photocurrent. Having uneven variance is not of any sig- nificant concern, though, since the method of weighted least squares was used for the regression (it could be a problem if an unweighted least squares fit were used).

0 0.1 0.2 0.3 0.4 0.5 0.6 0

.5

1

1.5

2

2.5

3

3.5

4

R etard i n g Vol tage (V )

P h

o t o c u

r r e n

t ( x

1 0

! 7

Å )

! = 6500 Å

Vs t o p= 0. 4 3 4 1 5 V

FIG. 3: Photocurrent vs. Vr for 6500 Å light. The equation for the fit line is: yfit = 5.14042×10−7 + (−1.18402×10−6) ν. The uncertainty in the y-intercept is σa = .1254×10−7 A and in the slope is σb = .3054 × 10−7 A/V .

In comparing Figure 3 to Figure 2, one sees that the basic shape of the data is the same, but that the linear drop occurs at a lower voltage with the longer wavelength light and that the photocurrent has a smaller amplitude due to the longer wavelength (and thus less energetic) light.

In Figure 4, the nine values of Vs are plotted against the frequency of light to which they correspond (where ν = c/λ and c = 3 × 108 m/s). Another weighted least squares regression was found using Equation 6 to find the parameters A and B used in Equation 5. This led allowed for the calculation of h and Φ. The error bars in Figure 3 are generated using the uncertainty in Vs, which is calculated using the equation

σVs = Vs

√ σ2a a2

+ σ2b b2

(14)

See the error analysis section for the derivation of this error propagation equation. Also see that section for an

4

0 2 4 6 8 10 12

−1

0

1

2

3

4

Frequ en cy (1 0 e14 H z)

S t o p

p in

g V

o lt

a g e

( V

)

FIG. 4: Stopping Voltage vs. Frequency. The equation for the fit line is yfit = −1.51059 + (4.20866 × 10−6) ν. The uncertainty in the y-intercept (found using Equation 7) is σA = .04609 V and in the slope is σB = 8.85979 × 10−17 V/Hz. The calculated R2 value for the fit line was .999999.

explanation of how the R2 value shown in Figure 4 was calculated.

The calculated parameters from the weighted least squares regression in Figure 4 were A = −1.51059 ± .046087 V and B = (4.20866 × 10−15 ± .088598) × 10−15 V/Hz. Plugging these values into Equation 5, Planck’s constant is found to be h = (−6.743 ± .142) × 10−34 J s and the work function of the metal is found to be (2.419 ± .0737) × 10−19 J (or 1.5106 ± .0461 eV ).

ERROR ANALYSIS AND DISCUSSION

The uncertainty in the results for Planck’s constant and the work function of the metal are found by propa- gating the error in Vs using Equation 7. Those Vs uncer- tainties, though, need an explanation as to their deriva- tion from the values of σa and σb, which were also found using Equation 7. To find σVs , one begins with Equation 9. The uncertainty can be thought of as analogous to a small change in a variable, and thus one can treat σx values as analogous to the differential dx. Then, consid- ering a small change in Vs in Equation 9, the quotient rule is applied:

dVs = −b da + a db

b2 . (15)

Now, divide Equation 11 by Vs = −a/b and simplify to find

dVs Vs

=

( −da

b +

a

b2 db

) −b a

dVs Vs

= da

a − db

b . (16)

Now, one makes use of the standard formula for the propagation of error in a sum (if z = x + y, then

σz = √ σ2x + σ

2 y. Substituting Equation 12 into this for-

mula and converting from differentials back to standard deviations, one finds

σVs Vs

=

√(σa a

)2 +

( −σb b

)2 , (17)

and solving for σVs the asserted relation is obtained:

σVs = Vs

√ σ2a a2

+ σ2b b2 . (18)

In addition to examining small the standard deviations found in the final fit parameters A and B, calculating the coefficient of determination, R2, for the fit line in Figure 4 indicates the high goodness of fit achieved. The coef- ficient of determination was again calculated “by hand” in Matlab. The equation used was

R2 = 1 − SSerr SStot

, (19)

NOTES 5

where

SStot = ∑

(Vs,i − Vs)2 (20)

SSreg = ∑

(Vs,i − yfit,i)2 (21)

where Vs is the mean of the Vs values. The obtained R2 value of .999999 indicates that the stopping voltage values varied very linearly with frequency.

CONCLUSION

This experiment demonstrated that Planck’s constant can be experimentally determined using the photoelec- tric effect. The value obtained was very close to the ac- cepted value; h = (−6.743 ± .142) × 10−34 Js represents a 1.77% error and the accepted value, 6.626 × 10−34 Js falls within the calculated uncertainty of the experimen- tal result. Thus, it may be concluded that the simulation program accurately models the photoelectric effect with some, but not a problematic amount, of error.

One of the largest sources of error in the data produced

by this program is likely the heavily fluctuating back- ground current. Were the experiment to be repeated, the background current measurement procedure could be al- tered slightly such that background is measured for every wavelength, in case the fluctuations actually do depend appreciably on wavelength. Given the heteroskedacity observed in the photocurrent error as retarding voltage increased, a similar effect might reasonably be observed with changing wavelength in the background current.

Otherwise, the highly accurate experimental value of Planck’s constant suggests that the determined value of the work function, Φ = (2.4193±.0737)×10−19 J, is very probably quite close to the “real” value of Φ programmed into the simulation.

Notes

1“PHOTO-02.exe”, Portland State University Physics Depart- ment, SB2, Room 201.

2Bevington and Robinson, Chapter 6, ”Least-Squares Fit to a Straight Line,” Data Reduction and Error Analysis for the Physical Sciences, 2nd ed., (McGraw-Hill: 1992).