Undergraduate OP03 Lab Report

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A written report for the OP03 undergraduate (First year) physics lab.
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  OP03 Written Report Grating Monochromator by Samuel Gillespie This experiment aimed to determined the Rydberg's constant experimentally, and found it tohave a value of  1.0970 x 10 7 ± 0.046%. This result is consistent with the theory obtained fromthe “Bohr model”. Differentiating between the R  ∞ (mass of nucleus >> mass of electron) andR  H (precise masses used) was unsuccessful, however, as the uncertainty in the obtained resultencompassed both values. 1 Introduction The “Bohr model” is an early approximation of an atom, and consists of a fixed, positivelycharged nucleus surrounded by electrons travelling around it in a circular orbit. From thismodel is it possible to predict the emission spectrum of a hydrogen atom, and it can be shownthat the emitted spectral lines depend on the following 1 :  Equation 1.1  This first value is called the “Rydberg's constant” and can be obtained from the Bohr model if the proton is taken to be infinitely larger than the orbiting electron.The second value is the Rydberg's constant for a nucleus of mass M. When then mass of thenucleus is m  p (as is such in the case of hydrogen) we are left with this equation:  Equation 1.2 The aim of this experiment is an attempt to validate this equation and test its accuracy. It isimportant to test this model, as it can be applied to other aspects of physics to achieve usefulresults. For example, it is possible to use the emission spectrum of atoms to determine thespeed of stars moving through the universe. This is achieved by observing the emissionspectrum given out by the moving star, and comparing it with a spectrum of known values. Bycomparing the known values with the corresponding values found the red-shift can bedetermined. Using this value an observer can then work out the speed that the body is travellingrelative to themselves.As hydrogen is found in abundance in main stage stars, knowing accurate spectra is veryuseful. From just the hydrogen spectra it is possible to determine the velocity of all main stagestars. 1Derivation can be found in the OP03 laboratory script 1  R  H  = ( 14 πϵ 0 ) 2 e 4 4 π c ℏ 3  xm e m  p m e + m  p  R ∞ = ( 14 πϵ 0 ) 2 m e e 4 4 π c ℏ 3  2 Experimental arrangement To find the wavelengths of light emitted by excited hydrogen it is possible to use a Czerny-Turner monochromator. To set up the monochromator the light source must first be focusedonto the entrance slit of the system. The light is then internally reflected through the systemtowards a collimating mirror. This produces collimated light, which is reflected further downthe monochromator and towards the reflection grating. This grating causes the light to diffract,with each wavelength of light being diffracted by varying amounts. When focused, this lightwill create bands of colour (called spectral lines) against a dark background. This focusing isachieved by, once again, reflecting the light on a concave mirror and reflecting the light to theeyepiece. If positioned correctly, the focus of the light will be directly on the eyepiece. This set-up can be seen in   Figure 2.1. As is seen in the diagram, after hitting the grating the different wavelengths of light will diffract by varying amounts, causing constructive interference at different angles leaving the grating.Thus they will focus at different places at the exit slit and rotating the grating will showdifferent wavelengths of light in the centre.To utilise this equipment the apparatus must be correctly set up. As previously stated, this firstrequires ensuring that the entrance slit is positioned so that light passing through it will becollimated by the collimating mirror. This is achieved by ensuring that (when the grating is setto reflect the light straight back) the image of the slit and the slit itself creates no parallax. Thisinfers that the image is lying directly on top of the object, and the focal point lines up with theslit.2  Figure 2.1: Czerny-Turner Monochromator   The best image on the cross-wires is achieved when the light from the light source is focuseddirectly on the slit with a magnification of one. To accomplish this, the light source must be positioned a distance two times the focal length away from the lens. The entrance slit must thenalso lie two focal lengths from the focusing lens (  Figure 2.2 ). This will thus create a totaldistance of four focal points between the slit and the light source. The exact placement is notnecessary though, as it just improves the image quality, helping to get more reliable results.The cross-wires must then also be positioned so that the light leaving the monochromator focuses on them. This will ensure the spectral lines are as thin as possible (sharp image) so thatit will be easier to determine when they appear centred on the cross-wires. It also ensure thereis no parallax when viewing image which would create a larger random error in the acquiredresults.After completing these steps, the apparatus is prepared and measurements can now be taken. Inthis case, an eyepiece is used to magnify the cross-wires and the image of the slit (and thusspectral lines). This enables the spectral lines to be seen against a black background, with thecross-wires also in view. From this point it is easy to modify the rotation of the grating anddetermine when a particular spectral line lies on the centre of the cross-wires. 3 Measurements Before the wavelengths of a particular light source can be observed with this apparatus it mustfirst be calibrated. The monochromator only measures the displacement of the grating inside it,and so these values must be converted into a wavelength to acquire the results desired. This canonly be done if an equation is known to convert between the two values, and discovering thisequation can be done by comparing the displacements found when viewing known wavelengthsof a spectrum.A mercury lamp emits a spectrum composing of many spectral lines (  Figure 3.1 ), of which thewavelengths are well defined. By plotting the values shown on the micrometer against thevalues of these known wavelengths it will be possible to fit the data to a line which can thenlater be used convert readings off the micrometer to wavelengths.3  Figure 2.2: Apparatus arrangement   To ensure accuracy, and to reduce random errors, measurements for each wavelength weretaken four times. Furthermore, when placing the line onto the cross-wires it was important toensure that the line is always approached from the same side. This is to stop the error thatwould be caused by the slight backlash in the mechanisms of the micrometer.After completing the calibration measurements (seeAppendix A) the mercury light wasreplaced with a capillary discharge tube that emitted the hydrogen spectrum. The light was positioned so that, when it passed through the lens, it focused onto the slit. The rest of theapparatus (excluding the eyepiece) remained fixed to ensure the calibration stayed valid.Once the light source had been switched, a spectrum of many spectral lines was found.However, only the four Bohr emissions in the visible spectrum were desired (the rest aremolecular emission). By using the Bohr model to estimate the location of the line, the estimatedreading on the micrometer could be found which would help distinguish between the manylines allowing the correct values to be measured.Once again, each reading was measured four times and the results were recorded in a table (seeAppendix B). 4 Analysis and results After gathering all the measurements needed the calibration results were analysed and a fittingcurve was produced.Upon plotting the points it appeared as though a straight line curve would satisfy the ratio between the grating displacement and the wavelength of light focused on the centre of thecross-wires. However, upon inspecting the residuals of this plot there is a noticeable curvaturein the fit of the them (Figure 4.1). Due to of this, a linear fit may not be the best options for thevalues recorded.4  Figure 3.1: Mercury Spectral Lines (From lab script) Figure 4.1: Residuals of linear fit  400 500 600 700-0.000010.000000.000010.00002    R  e  g  u   l  a  r   R  e  s   i   d  u  a   l  s  o   f  z Wavelength (nm) Regular Residual of Results zPolynomial Fit Regular Residual of Results z  Figure 4.2: Residuals of polynomial fit  400 500 600 700-0.000010.000000.000010.00002    R  e  g  u   l  a  r   R  e  s   i   d  u  a   l  s  o   f  z Wavelength (nm) Regular Residual of Results zPolynomial Fit Regular Residual of Results z
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