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

 

Abstract

In this experiment we will investigate the characteristics of the diffraction grating (Dispersion and Resolving Power), determine the separation of the sodium D-lines and investigate the atomic hydrogen spectrum and confirm the relation between Balmer series and quantum theory of hydrogen.

Introduction

A diffraction grating is a plane of glass containing a large number N of slits( of order of 10^⁴ slits per centimeter), and the distance d between two  successive slits is given by the width of the grating plate divided by N, which represents the reciprocal of the number of silts in one centimeter. The slits make  diffraction of the light falling on it and the distance between the fringes is relatively large for that the angle theta ɵ is large, and it is not possible to make an approximation, as in the Young double-slit experiment, and the distance between the Fringes that are produced on the screen is equal. The device Which use to measure the wavelength λ of light containing of the diffraction grating is called the Diffraction Grating Spectrometer.

When a beam of light incident on the grating, a set of fringes appear on the other side of the grating. Two fringes for each diffraction order appear symmetrically positioned on the two sides of the central bright fringe (see Figure 1). The angular position of the fringe of order m is given by equation 1. For a fixed direction of incidence, the direction of each principal maximum varies with wavelength. For orders therefore, the grating separates different wavelengths of light present in the incident beam, a feature that accounts for its usefulness in wavelength measurement and spectral analysis. spectral lines with longer wavelengths appear at larger angles and the angular separation between two different colors increases as the d-spacing of the grating decreases according to equation 1. As a dispersing element, the grating is superior to a prism in several ways.

The minimum of intensity, which determines the half-width (see figure 2), occurs when the additional path difference between each pair of rays is half of a wavelength(figure 3).

In a similar manner, we can consider that the additional path difference corresponding to the pair of rays emerging from the two slits at position x away in the same direction from the first slit and the central slit has the same value (this requires that the width of the grating is much smaller than the distance to the collecting screen). Thus, we conclude that equation (2) is the condition for producing a minimum intensity at  ɵ + Δɵ as a consequence of destructive interference between all rays emerging from the grating. Since the width of the fringe is small, the last equation reduces to the form equation 3.

There are some properties or characteristic of the diffraction grating which determine the efficiency of it.The most important of these characteristics are:

DISPERSION OF A GRATING

Higher diffraction orders grow less intense as they fall more and more under the constraining diffraction envelope. On the other hand, wavelengths within an order are better separated as their order increases. This property is precisely described by the angular dispersion D, defined by equation 4._[1]

In other wards it’s the ability of the grating to separate two wavelengths differing by Δλ to give diffraction peaks separated by an angular displacement Δθ.

RESOLUTION OF A GRATING

Increased dispersion or spread of wavelengths does not by itself make neighboring wavelengths appear more distinctly, unless the peaks are themselves sharp enough. The latter property describes the resolution of the recorded spectrum. By the resolution of a grating, we mean its ability to produce distinct peaks for closely spaced wavelengths in a particular order._[1]

To resolve two diffraction lines corresponding to two wavelengths differing by Δλ the angular separation Δɵ between these two lines should be large enough to detect. According to Rayleigh’s criterion, this separation must be at least equal to half the width of the line. Thus, high resolving power requires narrow diffraction lines. The resolving power is defined as the ratio of the average wavelength to the difference in wavelength  (equation 5).

According to Rayleigh’s criterion, the angular separation is given by equation (3). Thus, we obtain for the resolving power by equation 6.

Balmer Series

The hydrogen atom is the simplest quantum-mechanical system. It consists of an electron bound, due to the Coulomb force, to a proton. The total energy cannot have any value, but that the system is found in one of a discrete set of energy levels, or states. Transitions of the system between these states may occur. Such transitions must satisfy the basic conservation laws of electric charge, energy, momentum, angular momentum, and the other relevant symmetries of nature.

Transition from a higher energy state to a state with less energy can occur for an isolated system. During such spontaneous transitions, a quantum of radiation, or one or more particles, can be emitted, which will carry away the energy lost by the system. By observing the quanta of radiation, or the particles emitted during such transitions, we gain information on the energy levels involved. The typical example is optical spectroscopy, which consists of the accurate determination of the energy of the light quanta emitted by atoms.

The idea of energy levels and their structure for the hydrogen atom was first introduced by Niels Bohr in 1913. However, a complete theoretical interpretation had to wait until the introduction of the Schrodinger equation in 1926. We will use the Bohr theory to predict the hydrogen energy levels, because it is so simple. even though it assigns the incorrect angular momentum to the states.

The postulates of the Bohr theory are that the electron is bound in a circular orbit around the nucleus such that the angular momentum is quantized in integral units of Planck's constant (divided by 2π) and that the electron in this orbit does not radiate energy unless a transition to a different orbit occurs. We can then calculate the radii of these orbits and the total energy of the system, potential plus kinetic energy of the electron (equation 10).

The energy levels of the hydrogen atom can be represented by (Figure 4). However, the lines observed in the spectrum correspond to transitions between these levels; this is shown in (Figure 5) where arrows have been drawn for all possible transitions. The energy of a line is given by (equ. 9)

Since the frequency of the radiation is connected to the energy of each quantum through (equ. 11).

From Eq. (9) (or from Fig. 4) we note that the spectral lines of hydrogen will form groups depending on the final state of the transition, and that within these groups many common regularities will exist, for example if (n=2) we will obtain the equ. (7) and all lines fall in the visible light ; they form the (so-called) Balmer series.

Methodology

1. Focus the viewing telescope for parallel rays on a distant object.
2. Adjust the collimator lens LC for parallel beam by focusing the slit without the grating in position.
3. Place the sodium source and focusing lens on position, turn the source on, and make sure you have good focusing and alignment. Then align the cross hairs with the slit.
4. Make sure that the position of the telescope is at zero to within the precision of the instrument.
5. Place the best grating you have (with the greatest number of rulings) in position for normal incidence and align it so that its lines are parallel to the crosshairs. Make sure not to rotate the
crosshairs in the process.

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6. Measure and record the angular position of the firs order diffraction line at both sides of the
central peak. Repeat the measurement for reproducibility. This position represents the average position of the sodium D-doublet since the doublet is not resolved in first order.
7. Find the second order diffraction peak and make sure that the doublet is now resolved. You can narrow the slit for better results. Record the positions of the two lines at both sides of the central peak. Then turn off the source.
8. Place the hydrogen source in place and recheck for alignment and focusing.
9. Record the positions of the spectral lines at the two sides of the central peak. Repeat the measurements for reproducibility. Then turn off the source.
10. Repeat steps (8) and (9) with the mercury lamp.

Formalism

Data Analysis

Part 1: D-sodium source: N = 600 / 1mm

m	ɵleft	ɵright	∆ɵ	∆ɵ/2	d (m)	λ (Åm)
1	22.605	44	21.395	10.6975	1.66*10-6	5890
2	41.49	41.475	41.4825	20.74125	1.66*10-6	5895.9

d_th = 1/N = (1/600)* 10^^-3 m

λ₁ = 5890 Åm , λ₂ = 5895 Åm ➨    λ_ave = 5892.95 Åm

d_exp = m λ_ave / sinɵ =  1.79*10^^-6 m

Percentage Error = │(1.79 – 1.66) / 1.66│*100% = 7.8%

λ₁ = d sinɵ / m = (1.79*10^^-6 * sin(41.49)) /2 = 5930 Åm

λ₂ = (1.79*10^^-6 sin(41.475)) /2 = 5928 Åm

P.E (λ₁) = │(5930-5890)/5890│*100% = 0.6%

P.E(λ₂) = │(5928 – 5895.9) / 5895.9│*100% = 0.5%

Part 2: Hydrogen Source 

N = 600 , d = (1 / 600 ) *10^^-3 = 1.66 * 10^^-6 m.

Color	ɵR	ɵL	ɵ	λ	n	1/λ	1/n2	λ 0 nm
Violet	17.808	48.631	30.823	4.395	5	0.227531	0.04	438.07
Turquoise	16.008	50.309	34.301	4.89	4	0.204499	0.0626	486.133
Red	9.161	56.622	47.461	6.62	3	0.151057	0.111111	656.272

R = Slope = 0.926 *10^⁷ Rydberg`s constant

R₀₀ = 1.0974 *10^⁷

PE = │(0.926 – 1.0974)  / 1.0974│*100% = 15.6%1-

1 / λ = R₀₀ (1/4 – 1/n^²)

0.271 = R₀₀ (1/4)

R₀₀ = 1.084* 10^^-7

PE = │(1.084 – 1.0974) / 1.0974│*100% = 1.22%

Last Squared Method:

Conclusion

1-We use the spectrometer which contains the diffraction grating to calculate the wavelength of a monochromatic light by using Bragg`s law(equ. 1).

2-We use it also to investigate the spectral lines of the hydrogen atom and confirm the relation between Balmer series and the quantization theory of hydrogen(Bohr hypothesis), we do this by calculating the wavelength for each color by using equ.1 and calculating the corresponding  principle quantum number n by using Balmer series then we find the Rydberg’s constant which represent the slope of the relation the inverse of the wavelength versus 1/n². We find that our result in an good agreement with the theory with percentage error 1.22% the result so close to that expected.

3-We use this device to find the separation between the d-lines of the sodium by measuring ɵ for each line from the right and the lift then we take the average of them. After that, we use equ.1 to find λ for each line and we take the average. Finally, after we calculate the resolving power from equ.5 we calculate the smallest separation of the d-lines from equ.6.   our result was in an acceptable agreement with the theory with percentage error of the separation about 66.2%.

References

1-Third Edition, Introduction to Optics, FRANK L. PEDROTTI, S.J.
LENO M. PEDROTTI, LENO S. PEDROTTI.

2- EXPERIlVIENTS IN MODERN PHYSICS, Second Edition, Adrian C. Melissinos.

3- ADVANCED PRACTICAL PHYSICS (0352311), Sami Mahmood, The University of Jordan.