Investigating the Absorption of Radiation in Matter using G-M Counter

 Abstract

In this experiment, we shall study the Kerr electro-optic effect and determine the Kerr constant for Nitrobenzene using a source of light, polarizers and the Kerr cell which consists of a glass container filled with an optical active material ( which is nitrobenzene in this case) and two parallel plates immerged in the cell connected to voltage difference source and finally a photocell to convert the photons to an electric signals.

Introduction

Nonlinear electro-optic effects result from the application of DC or low-frequency electric field to a medium. These effects cause some modulations to the light. By light modulation, we mean the modification of the amplitude, frequency, phase, polarization, or direction of a light wave. One purpose of modulation is to render the wave capable of carrying information.

One of these effects is the Kerr effect, which discovered in 1875 by Kerr. He found that when a plate of glass is putted in a strong electric field it becomes doubly refracting. This effect occurs in solid, liquids and even in gases.

In this experiment and to study this phenomenon we will use Kerr cell which consists of a glass container filled with an optical active material (Which is nitrobenzene in this case) and two parallel plates immerged in the cell connected to voltage difference source. What does the optical active material mean? It is the material that interacts with light and change its properties.

When we applied strong electric field on the substance it becomes doubly refractive. What do we mean by doubly refracting material? Those materials in which the speed of light and then the refractive index depend on the direction of the propagation of the light wave. This occurs because the molecules of the organic liquids are not spherically symmetric (polar molecules) and when an external electric field E is applied, these molecules are aligned in the direction of E . A wave of light propagating in such an aligned medium finds it inhomogeneous in that its susceptibility or polarizability is different for different directions of propagation.

What is the relation between susceptibility and polarizability?

Susceptibility indicates the degree of polarization of the dielectric material in response to an applied electric field. The greater the susceptibility the greater the ability of a material to polarize.

When a light wave passes through a medium, its velocity will be smaller than its speed in free space, which is given by the relation:

where m is the permeability of the medium, here considered equal to m₀, and ε is the permittivity of the medium.

Since the speed of light passing through a dielectric material depending on the susceptibility of the medium , then the permittivity becomes:

where ε₀ is the permittivity of free space, X is the susceptibility of the medium in the direction of the electric field vector of the propagating wave.

By substituting the second equation in the first one we get:

Which  shows the dependence of the velocity of light in a medium on the susceptibility of that medium.

Now  To determine the effect on a light wave of frequency f (angular frequency w =2p f ) and wavelength λ, propagating through the medium, we first determine k, the magnitude of the propagation vector, which is given by:

The phase difference between the point of entry of the wave into the medium and the point of exit is given by:

where L is the path length. When we substitute the value of K and then the value of v we get:

If the medium were doubly refracting, the value of φ could vary and would depend on the direction of the electric field vector. So we have two values X₁ and X₂ depending on the axis.

Experimentally and in our case the applied electric field is along the y-axis, the direction of propagation of the light wave is along the z-axis. This gives rise to two principal axes along the x and y directions.

The incoming light wave is linearly polarized in the xy plane at an angle of 45° to the x-axis. This allows the electric field vector of the wave to be decomposed into two equal components: one along the x-axis, and the other along the y-axis.

As the wave traveling through the Kerr cell, a phase difference occurs between the x and y components ( Dφ = φ -φ ) , The phase difference  Dφ  is experimentally found to be proportional to the square of the applied electric field and to the path length in the medium. 

where K is the Kerr constant.

The out electric field from the cell is equal to:

Where E_o is the amplitude of the electric field vector, and :

The intensity of light is proportional to the square electric field vector (Ι_out):

Combining the equation above and the equation of Kerr cell we get:

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The applied electric field between capacitor’s plates equal to the voltage divided by the distance between the parallel plates.

Experimentally we can record the current for each voltage, and to determine Kerr constant for the nitrobenzene we plot Sinˉ1 (Ιout/Ιo) vs V^², the slope will equal to πKL/d^², then:

Methodology

The optical system is arranged as shown in the figure above.

The lamp lens is moved to focus the light onto the iris. Lens 1 is placed such that a parallel beam of light results. The polarizer is arranged at 45° to the horizontal. The slit is arranged to only allow light to pass between the plates of the Kerr cell. The analyzer is arranged perpendicular to the polarizer so that no light passes through with no voltage applied to the cell.

The photocell is connected to the galvanometer and the high voltage supply terminals are connected to the Kerr cell. The voltmeter is connected across the power supply and set to the 0-2500 V DC range.

The experimental procedure consists of taking the output light intensity readings as a function of the applied voltage.

1. Vary the voltage between 1300 and 400 volts taking readings every 50 volts.

2. It is advisable to start at the higher voltage of 1300 V and then reduce your voltage in steps of 50 volts until you reach 400 V. This yields more stable voltage readings.

3. Note: the current is unstable below 400 V.

4. It will be noticed that the output intensity increases with increasing voltage up to a maximum value, and then decreases as the voltage is increased.

5. Take many readings around the maximum value of the intensity. The voltage must not exceed 1500 V under no circumstance.

6. When the voltage is applied for the first time, it is seen to rise slowly.

7. Wait until the readings stabilize before taking intensity measurements.

8. The experimental run should be repeated a few times to ensure reproducibility of the results.

Data Analysis

I₀ = 11.8mA (Intensity with the polarizer and analyzer parallel)

Volts (v)	I(10^-6) A	v^2	I/I0	(I/I0)^1/2	Arcsin((I/I0)^0.5)
1300	9.2	1690000	0.779661017	0.882984154	1.08218202
1250	10.3	1562500	0.872881356	0.934281197	1.206237642
1200	11.2	1440000	0.949152542	0.974244601	1.343346457
1150	11.7	1322500	0.991525424	0.995753696	1.478608342
1100	11.8	1210000	1	1	1.570796327
1050	11.4	1102500	0.966101695	0.982904723	1.385625011
1000	10.7	1000000	0.906779661	0.952249789	1.260521806
950	9.8	902500	0.830508475	0.911322377	1.146484771
900	8.6	810000	0.728813559	0.85370578	1.023060481
850	7.3	722500	0.618644068	0.786539298	0.905184814
800	6.1	640000	0.516949153	0.718991761	0.802350564
750	4.9	562500	0.415254237	0.644402233	0.70024131
700	4	490000	0.338983051	0.58222251	0.621459639
650	3	422500	0.254237288	0.504219484	0.528477908
600	2.3	360000	0.194915254	0.441492077	0.457260911
550	1.7	302500	0.144067797	0.379562639	0.389323512
500	1.3	250000	0.110169492	0.331917899	0.338336012
450	1	202500	0.084745763	0.291111255	0.295388196

Figure 1: The current as a function of voltage.

We noticed that the curve reached his maximum value at V=1100 with I=11.8

Figure2: V squared versus Arcsine ((I\Io) ^1\2)

Slope= 8*10 ^-7 rad/ V^²

From  K=(slope*d²)/πL

Where:

d= 0.5*10^-3 m

L=4*10^-2 m

K=1.591549431*10^-12 (rad.m)/V^²

v^2	Arcsine((I\I₀)^1\2)	x^2	Yi-a	x*y	yi-a-bx	(yi-a-bx)^2
1690000	1.08218202	2.8561E+12	0.803482	1828888	-0.54851798	0.300871974
1562500	1.206237642	2.44141E+12	0.927538	1884746	-0.322462358	0.103981972
1440000	1.343346457	2.0736E+12	1.064646	1934419	-0.087353543	0.007630641
1322500	1.478608342	1.74901E+12	1.199908	1955460	0.141908342	0.020137978
1210000	1.570796327	1.4641E+12	1.292096	1900664	0.324096327	0.105038429
1102500	1.385625011	1.21551E+12	1.106925	1527652	0.224925011	0.050591261
1000000	1.260521806	1E+12	0.981822	1260522	0.181821806	0.033059169
902500	1.146484771	8.14506E+11	0.867785	1034703	0.145784771	0.021253199
810000	1.023060481	6.561E+11	0.74436	828679	0.096360481	0.009285342
722500	0.905184814	5.22006E+11	0.626485	653996	0.048484814	0.002350777
640000	0.802350564	4.096E+11	0.523651	513504.4	0.011650564	0.000135736
562500	0.70024131	3.16406E+11	0.421541	393885.7	-0.02845869	0.000809897
490000	0.621459639	2.401E+11	0.34276	304515.2	-0.049240361	0.002424613
422500	0.528477908	1.78506E+11	0.249778	223281.9	-0.088222092	0.007783138
360000	0.457260911	1.296E+11	0.178561	164613.9	-0.109439089	0.011976914
302500	0.389323512	91506250000	0.110624	117770.4	-0.131376488	0.017259782
250000	0.338336012	62500000000	0.059636	84584	-0.140363988	0.019702049
202500	0.295388196	41006250000	0.016688	59816.11	-0.145311804	0.02111552

To obtain the error of K (𝝙K) we will calculate the standard deviation.

S=(∑(𝒚𝒊−𝒃𝒙𝒊−𝒂) ²)^ ^0.5/ (𝑵−𝟐) ^^0.5

Where a is the intercept and b is the slope

a=0.2787

b=8*10^-7

N=16

We will use the error of the slope:

= 1.5406144*10^-7

 

𝞓𝑲 𝑲 = 𝜟𝑺𝒍𝒐𝒑𝒆 /𝑺𝒍𝒐𝒑𝒆

 𝝙k=K * 𝜟𝑺𝒍𝒐𝒑𝒆 /𝑺𝒍𝒐𝒑

 

𝝙k=3.06495506*10^-27 m/ volt

Conclusion

v  When we applied an electric field on a dielectric material, its molecules will  be aligned with the direction of electric field, such aligned media would exhibit different indices of refraction, and hence become doubly refracting.

v  Kerr constant depends mainly on the wavelength and temperature so we should use a monochromatic wavelength in study this effect, but in our case the dependence of nitrobenzene material  on the wavelength and temperature is not very strong so we can ignore these factors and use a non-monochromatic light source.

v  We use the first lens in the experiment setup to make the light rays parallel and when we parallel rays it means that light waves are plane waves.

v  If we know the Kerr constant, we will be able to determine the type of the substance, because every substance has a specific value of Kerr constant.

 References

1- Practical physics (0342411) manual.

2- Third Edition Introduction to Optics, PEDROTTI.