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Visible Spectrometer PHY4803L — Advanced Physics Laboratory∗ University of Florida Department of Physics (Dated: September 15, 2010) The Balmer spectral lines from a hydrogen discharge lamp are observed with a transmission grating spectrometer and analyzed to obtain the Rydberg constant. Wavelength calibration is achieved by measuring diﬀraction angles for spectral lines of “known” wavelengths from a mercury and helium discharge tube and ﬁtting this data to a grating equation. The wavelengths of the hydrogen lines are then determined and ﬁt to the Rydberg formula. Basic spectroscopic and statistical analysis techniques are employed. 1. THEORY is called the inﬁnite mass Rydberg or simply the Ryd- berg. Although RH and R∞ diﬀer by less than 0.1%, with 1.1. The Hydrogen Spectrum care, the measurements you make should be just accurate enough to distinguish the diﬀerence between them. Sets of wavelengths (series) are categorized by the Both the semi-classical Bohr model (circular electron quantum number nf of the lower level of the transition. orbit with quantized angular momentum) and the solu- The Lyman series is obtained for nf = 1, ni = 2, 3, ... and tions to the spinless Schroedinger equation for the hy- is in the ultraviolet part of the spectrum; the Balmer se- drogen atom lead to discrete energy levels that can be ries corresponds to nf = 2, ni = 3, 4, ... and is in the expressed in terms of a principal quantum number n visible; the Paschen series, for nf = 3, is in the infrared; µe4 etc. En = − 2 h2 n2 (1) 8 0 1.2. The Diﬀraction Grating where e is the electron charge, 0 is the permittivity of free space, h is Planck’s constant, and µ is the reduced mass of the electron-proton me -mp system, me mp transmission µ= (2) grating me + mp collimator central beam A photon is emitted when the electron makes a transi- θi θr slit diffracted grating beam tion from a higher energy level to a lower level with the normal telescope photon carrying away the excess energy ∆E = Eni −Enf . Because the energy of a photon and its wavelength λ in vacuum are related by λ = hc/∆E, Eq. 1 predicts the relation FIG. 1: Top view of a transmission grating spectrometer. Note that the incident angle θi and the diﬀraction angle θr 1 1 1 are relative to the grating normal. = RH − 2 (3) λ n2f ni Wavelength measurements in this experiment are where based on the interference of a large number of waves scattered from the grooves of a transmission grating illu- µe4 minated by incident plane waves of various wavelengths. RH = (4) 8 2 h3 c 0 The geometry is shown in Fig. 1. Exercise 1 Draw a ﬁgure showing adjacent grating is called the reduced mass Rydberg. If the nucleus were in- grooves (spaced d apart) and show that constructive in- ﬁnitely heavy, the reduced mass µ becomes me , the mass terference occurs for of the electron. The combination of physical constants mλ = d(sin θr − sin θi ) (6) me e4 R∞ = 2 3 (5) where the incident angle θi and the diﬀraction angle θr 8 0h c are measured relative to the grating normal and have the same sign when they lie on opposite sides of the grating normal (as in Fig. 1). The diﬀraction order m is a pos- ∗ Electronic address: deserio@phys.uﬂ.edu; URL: http://www. itive integer (θr > θi ) negative integer (θr < θi ) or zero phys.ufl.edu/courses/phy4803L (θr = θi ). Visible Spectroscopy Advanced Physics Laboratory 1.3. Dispersion and Resolution Two important spectrometer properties are dispersion and resolution. Dispersion is a measure of the rate of change of spectral line position with λ. With higher dis- persion, the spectral lines are more separated from each other or more spread out. With our spectrometer, spec- tral line positions are measured as an angle θr and dis- persion is best represented by the value of dθr /dλ. Exercise 2 (a) Obtain an expression for dθr /dλ from the grating equation (Eq. 6) in terms of d, m, and θr . Show that it increases for larger order number m, smaller d, and larger θr . (b) For a given λ, m, and d, will the dis- persion increase, decrease, or remain the same as θi in- creases? Hint for part b: As θi increases, θr must change as well. Figure out how θr must change. Then from part (a) you will know how the dispersion will change. The end result is that the answer to part b will depend on the FIG. 2: Top: Many-slit diﬀraction pattern for a single wave- sign of m, i.e., to which side of the grating normal you length. Bottom: The thick line is the sum of two diﬀrac- are measuring. The dispersion will increase on one side tion patterns for two diﬀerent wavelengths separated by the and decrease on the other. You should derive and explain Rayleigh criterion. this dependence. Exercise 3 If θi were 25◦ and the grating had 600 lines/mm, ﬁnd all angles θr where a red line at 650 nm and a blue line at 450 nm should appear in ﬁrst order (m = ±1) or second order (m = ±2). Hint: You should get only seven angles not eight. Which one isn’t possible? Assume you made measurements at these seven angles with an uncertainty in θr of 4 minutes of arc (4/60◦ ) for all of them. Provide an analytic expres- sion for the propagated uncertainty in λ and determine its value for each measurement. Resolution describes the ability of the spectrometer to separate two nearby spectral lines. Imagine the wave- length separation between the two spectral lines becom- ing smaller. The observed lines will begin to overlap each other and at some small separation ∆λ the abil- ity to discern the lines as separate will be lost. Resolu- FIG. 3: The spectrometer for this experiment. The various tion is a measure of the smallest discernable ∆λ. When components are described in the text. the spectral linewidths are dominated by spectrometer settings, such as slit width and focusing quality, higher dispersion generally implies higher resolution. But the where N is the number of grooves illuminated, and m is narrowest linewidths are ultimately limited by the num- the order of diﬀraction in which they are observed. λ/∆λ ber of grating grooves illuminated. As the number of is called the resolving power. grooves contributing to the diﬀraction increases, the an- Fig. 3 shows the location of the various spectrometer gular width of the diﬀraction pattern (and hence the adjustment mechanisms. Familiarize yourself with them. minimum linewidth) decreases. (Large gratings are used Components in italics below will be referred to through- when high resolution is needed.) out the write-up. The Rayleigh criterion gives a generally accepted mini- Note the two sets of rotation adjustments (J and K) mum “just resolvable” ∆λ. The criterion is that the peak near the legs of the spectrometer. The upper one (K) of the diﬀraction pattern of one line is at the ﬁrst zero of aﬀects rotations of the telescope (D), and the lower one the diﬀraction pattern of the other. See Fig. 2. Diﬀrac- (J) aﬀects rotations of the table base (N) on which the tion theory can be used to show this condition occurs for actual table (C) (holding a prism or grating) is inserted. two lines of wavelengths λ ± ∆λ/2 when Not shown is a mounting post on the table. Both ro- λ tational motions are about the main spectrometer axis = mN (7) (Q) which is vertical and through the center of the in- ∆λ 2 Visible Spectroscopy Advanced Physics Laboratory ing) screw and rotate the lower (adjustment) screw to 30 20 10 tilt the optic axis up or down. When properly adjusted, 0 ﬁnger tighten both screws while maintaining the axis ori- entation. A grating mounted on a glass plate with a grating spac- ing d ≈ 1/600 mm will be used for the measurements. The grating should only be handled by the edges of the glass plate. Do not damage the grating by touching it. 150 130 140 1.4. Alignment Procedure ◦ FIG. 4: The angular reading is 133 9 . The 0 mark is just after the 133◦ line on the main scale. The 9 mark is directly This spectrometer is a semi-precision instrument that aligned with a mark on the main scale. can be damaged with improper use. Parts should not be removed for any reason without ﬁrst checking with an instructor. strument. Each rotation adjustment has a locking screw 1. Carefully take the spectrometer out into the hall- and a tangent screw. When the locking screw is loose, way and place it on a portable table or lab stool. the corresponding element (telescope or table base) can Look through the telescope at the bulletin board be rotated freely by hand. When tightened, the corre- near the Student Services oﬃces at the end of the sponding element can be rotated small amounts with ﬁne corridor. This is far enough to be an eﬀectively control using the tangent screw. There are two precision inﬁnite object distance. With both eyes open and machined circles (called divided circles) associated with the unaided eye focused on the distant object,1 si- these elements. The outer circle is rigidly attached to the multaneously focus on the cross hairs by sliding the telescope and has an angular scale from 0 to 360◦ in 0.5◦ eyepiece in or out and focus on the bulletin board increments. The adjacent inner circle is rigidly attached using the telescope focusing knob. The telescope to the table base and has two 0 to 30 (minutes of arc) should now be focused at inﬁnity and the cross hair vernier scales on opposite sides. image should be located at inﬁnity. Under these When measuring angles it is important to take read- conditions there should be no parallax between the ings at both vernier scales. Because of manufacturing bulletin board and cross hair images; as you move tolerances, the two readings will not always be exactly your eye slightly from side to side, the cross hairs 180◦ apart. By using them both, more accurate angular should not move relative to the image of the bul- measurements are obtained. See Fig. 4 for an example of letin board. If it does move, the bulletin board a reading at one vernier scale. image is not at the cross hair image and the tele- The table is mounted on a post that is inserted into scope and/or eyepiece focus still needs adjustment. the table base and locked into place with the table locking It may help to defocus the telescope and then read- screw (I). Three tilt adjustment screws (O) on the table just the eyepiece to focus only on the cross hairs allow the pitch and yaw of the table to be varied. (with relaxed eyes focused at inﬁnity) before try- Cross hairs inside the telescope are brought into focus ing to refocus the telescope. When both the cross by sliding the eyepiece (L) in or out. The telescope is hairs and bulletin board are focused and show no focused by rotating the telescope focusing ring (E). The parallax, bring the spectrometer back to the lab collimator (B) (on which the entrance slit (A) is located) bench. is adjusted for focus by sliding the inner collimator tube in or out. An index ring (F) on the collimator tube al- 2. Loosen the index ring, move it back against the en- lows the focusing position to be maintained once found. trance slit assembly and retighten it. You should The ring has a V-shaped protuberance that ﬁts into one now be able to grab the index ring to move the of the V-shaped cutouts spaced 90◦ apart on the outer collimator tube smoothly in and out and rotate it. collimator tube. Making registration with one of the V’s Place a incandescent bulb behind the entrance slit once the proper focusing and slit orientation is obtained, and line up the telescope to look into the colli- the ring is then tightened into place. This permits the mator. Looking through the telescope, move the slit orientation to be changed 90◦ without losing the fo- cus by sliding the tube out slightly, rotating it 90◦ , and pushing it back into the other V-shaped cutout. 1 An image viewed through an eyepiece can be located anywhere The telescope and collimator also have leveling screws (P) that allow their optical axes to tilt up or down. These from your near point (about 30 cm for most people) to inﬁnity and still be focused upon. Keeping the eye that is not looking adjustments are used in a special procedure to orient both through the eyepiece open and focused at inﬁnity (as best you optic axes in a common plane perpendicular to the main can under such conditions) helps ensure that the image viewed spectrometer axis. To use them, loosen the upper (lock- through the eyepiece will also be located at inﬁnity. 3 Visible Spectroscopy Advanced Physics Laboratory collimator tube in or out until the open slit is in angle. Center it vertically by adjusting the tele- sharp focus. Do not touch the telescope fo- scope leveling screws. If the focusing is correct, you cus ring. It must be left where it was from should be able to see a reﬂected cross hair image in the previous step. Adjust the entrance slit for the circle. If not, adjust the telescope focus to get a narrow width and orient it horizontally. Move the reﬂected cross hairs in focus with no parallax the telescope slightly from side to side and verify- between the two cross hair images.2 Further adjust ing that the cross hair center moves parallel to the the telescope angle and the leveling screws to align slit. This guarantees the slit is horizontal. Don’t (overlap) the center of the two cross hair images. be concerned if the slit is slightly high or low rela- Overlapping the cross hairs and their reﬂection is tive to the cross hair center. Carefully—without called autocollimation. In this step, only the verti- moving the collimator tube—loosen the index cal alignment is important; it ensures the telescope ring completely. Then gently move it into one of axis is perpendicular to the rotation axis; you can the V-grooves and retighten it—again, without rotate the telescope from side to side to make sure moving the collimator tube. At the end of this the two cross hair centers overlap as they pass by step you should have a horizontal and sharp image one another. Achieving overlap both vertically and of the entrance slit. horizontally ensures the telescope axis is parallel to the grating normal. Since the grating normal was 3. Using the telescope leveling screws, get the cross made perpendicular to the main rotation axis in hairs centered on the narrow (and still horizon- step 6, this step makes the telescope axis perpen- tal) entrance slit. Tighten the leveling screws dicular as well. Tighten the leveling screws while while maintaining the vertical alignment. This en- maintaining the vertical overlap of the two sets of sures the telescope and collimator optical axes are cross hairs. aligned to one another but not necessarily perpen- dicular to the main rotation axis. 8. Loosen the table locking screw and carefully remove the table and grating, setting them aside without 4. Rotate the telescope until it makes an 80-90◦ angle upsetting the position of the grating relative to the with the collimator. table. Align the telescope and collimator to directly view the entrance slit. If necessary, adjust the tele- 5. The front face of the grating (actually the glass scope focus to focus the entrance slit. Adjust the the grating is mounted on) will reﬂect some of the collimator leveling screws so that the cross hair light incident on it. For this alignment procedure center is vertically aligned with the still horizontal the front surface of the grating will be used as a entrance slit, and then retighten them. This step mirror. Mount the grating on the table and tighten makes the collimator axis parallel to the telescope it under the clamp. Make sure that the grating is axis and thus also perpendicular to the main rota- well centered and in line with the mounting post tion axis. and the opposite tilt adjustment screw, bisecting the other two tilt adjustment screws as shown in 9. Reinstall the table/grating and go back to step 4 to Fig. 5. Adjust the table height to get the grating check and reﬁne your alignment. Check that after centered on the collimator. alignment is complete: (a) When the telescope eye- piece is focused on the cross hairs, any user would 6. Adjust the table base rotation angle as necessary then ﬁnd that the entrance slit is also in focus with- to see (through the telescope) the reﬂection of the out parallax relative to the cross hair image. (b) entrance slit oﬀ the grating glass. Tighten the table The entrance slit is oriented horizontally. (c) With and table base locking screws and adjust the table the telescope about 90◦ from the collimator and base rotation and the three table tilt adjustment with the grating glass used as a mirror, the reﬂec- screws to center the reﬂected image of the slit on tion of the slit is vertically aligned with the cross the cross hairs. This step guarantees that the grat- hairs. (d) With the telescope angle adjusted to the ing normal is perpendicular to the main rotation grating normal, the reﬂected cross hair image is in axis. Try to appreciate why this works! focus and vertically aligned with the direct cross 7. Move the telescope half-way toward the collimator hairs. so that the telescope is approximately perpendicu- lar to the grating. Shine a light into the side hole 10. Adjust the table rotation and the table base to get near the telescope eyepiece and then rotate the tele- an angle of incidence θi around 25-30◦ while en- scope until you can see light reﬂected oﬀ the grating glass. As you get close to being perpendicular, a partial circle of light will appear in the telescope. This illuminated circle will be as big as the ﬁeld of 2 Focusing the telescope on the reﬂected cross hair image while view but it will probably not be centered. Center it the eyepiece is focused on the actual cross hairs is another way horizontally by the ﬁne adjustment of the telescope to ensure the telescope is focused at inﬁnity. 4 Visible Spectroscopy Advanced Physics Laboratory The measurement of angular positions consists of lining up a feature with the cross hair center and recording tilt adjustment angular readings from both verniers. Do not make any subtractions or averages before recording. Label column screw holes grating headings A for one vernier and B for the other. A common spectroscopic technique for measuring un- post & known wavelengths from one source is to ﬁrst measure arm known wavelengths from some other sources. These ini- tial measurements are used to calibrate the spectrometer (determine constants such as the groove spacing d and the angle of incidence θi ) and are called calibration mea- surements. While it is perhaps a bit artiﬁcial, in this experiment the hydrogen wavelengths will be considered grating table unknown and those from any other sources will be con- sidered the known calibration wavelengths. It is recom- mended that both mercury and helium be used as cali- FIG. 5: Schematic top-view of the grating placement relative bration sources, but feel free to use other sources as you to the three table tilt screws. see ﬁt. The following measurement and analysis steps outline how this is done. suring that: (a) The grating height is centered on 14. Record the telescope angles An and Bn where the the telescope and collimator axes with the open end autocollimation signal from the grating reﬂection of the grating mount facing the collimator. (Other- occurred in step 11. This is a measure of the grating wise, the mount will block the spectral lines at large normal direction. angles.) (b) The vernier scales are roughly 90◦ to the collimator (and thus easy to view). Tighten the 15. Turn the entrance slit vertical and make it reason- table and table base locking screws. They should ably narrow (a few tenths of a millimeter). There not be touched again. are trade-oﬀs between the slit width and the abil- ity to see weak spectral features. The narrower the 11. In case the grating moved relative to the table, and slit, the sharper the lines, but they also get harder to accurately measure the location of the grating to see. normal, another autocollimation needs to be per- formed. Rotate the telescope to near normal inci- 16. Place a calibration source (e.g., helium or mercury dence on the grating glass and shine a light into the discharge lamp) just behind and nearly touching telescope side hole. Further adjust the telescope ro- the entrance slit. Adjust its position for maximum tation and the table tilt screws to ﬁnd and align the brightness while viewing a spectral line. Measure cross hairs with their reﬂected image. and record the angular reading Ai and Bi for the straight-through, zero order (all wavelengths) im- 12. With the slit still horizontal, place an incandescent age. This reading is a measure of the incidence light source behind the entrance slit. Rotate the direction. Note the “ghost” lines from imperfec- telescope to one side and the other and ﬁnd the tions in the grating. These ghost lines may also be “rainbows” on each side. Adjust the grating dis- seen on stronger spectral lines. Ignore them. persion plane using the inline table tilt adjustment screw (opposite the post in Fig. 5) so that the rain- 17. Make a table with columns for the color of the line bows on each side remain aligned with the cross and both vernier readings Ar and Br which would hairs. then be a measure of the diﬀraction angle. Record the readings for the brighter lines of the calibration 13. Repeat from step 11 until no further adjustments sources in all orders attainable on both sides of the are necessary. zero order image. At this point the spectrometer is ready for measure- 18. Place a hydrogen discharge lamp behind the en- ments. trance slit and adjust its position as for the calibra- tion lamps. The discharge should have a bright red section in the middle of the tube. If the discharge is 2. MEASUREMENTS all or mostly pink (less than a couple of centimeters of red in the middle), it is time to change the tube. Make sure the table base locking screw is tightened and Make a similar table of color, Ar , and Br for the ob- that you do not accidentally use the tangent screw. This servable lines of this spectrum. You should be able would cause the incidence angle to change and it should to see the Balmer lines corresponding to ni = 3, remain ﬁxed throughout the experiment. 4, 5, and 6 in several orders. They appear as a 5 Visible Spectroscopy Advanced Physics Laboratory violet (weak, sometimes extremely weak), blue vi- Equation 6 in terms of H-readings becomes: olet, blue green, and red. Feel free to use the video camera (if available) to see the weaker lines of hy- mλ = d [sin(Hr − Hn ) − sin(Hi − Hn )] (10) drogen. Just point it in where you put your eye and focus it. With the camera aperture fully open, the The values d, Hi , and Hn are constants in the ﬁt and all lines for ni = 6 and 7 (and perhaps higher) should three can be determined from a linear regression. be measurable in ﬁrst (and perhaps higher) order. C.Q. 1 (a) Use the trigonometric identity sin(a ± b) = CHECKPOINT: The procedure should be com- sin a cos b±cos a sin b, but only for the term sin(Hr −Hn ), plete through the prior step, and analysis should to show that Eq. 10 can be written: be complete through step 2. mλ = Dc sin Hr + Ds cos Hr + D0 (11) 19. Use the sodium lamp and the narrowest possible entrance slit. Observe the sodium doublet lines at where 589.0 and 589.6 nm in ﬁrst order (m = 1). They should be easily resolved — appearing as two sep- arate yellow lines. Now place the auxiliary slit Dc = d cos Hn (12) over the collimator objective and orient it verti- Ds = −d sin Hn cally. Slowly decrease its width. Since the light D0 = −d sin(Hi − Hn ) leaving the collimator is a parallel beam, as the aux- iliary slit is narrowed, less and less grating grooves Equation 11 is in the form of a linear regression of mλ on will be illuminated. Narrow the slit until the dou- the terms sin Hr , cos Hr , and a constant. The numerical blet is no longer resolved and appears as a single values for the three coeﬃcients: Dc , Ds , and D0 obtained line. Measure the auxiliary slit width at the point from the ﬁt can then be used to determine d, Hn and Hi . where the sodium lines are no longer resolved. De- (b) Show that d and Hn can be determined from termine how many grating grooves are illuminated. Do the lines then become resolvable when viewed in 2 2 d = Dc + Ds (13) second order? Discuss the signiﬁcance of this mini- experiment. Be quantitative. What is the resolving Hn = ATAN2(Dc , −Ds ) power of the spectrometer in ﬁrst order assuming diﬀraction limited performance when the auxiliary and that Hi can then be found using these values and slit is removed? Hi = − sin−1 (D0 /d) + Hn (14) 3. DATA ANALYSIS The ATAN2(x, y) inverse tangent function is available on Excel and guarantees the returned angle θ is in the 3.1. Calibration correct quadrant such that x = r cos θ and y = r sin θ (with r2 = x2 + y 2 ) will both be correctly signed. The ﬁrst step is to reduce each pair of angular readings A and B to a single value H by averaging the readings 2. Make side-by-side columns for sin Hr and cos Hr . for A and B ± 180◦ . Choose the sign in B ± 180◦ such Keep in mind that Excel’s trig functions need ar- that this term is near that of the A reading. For example, guments in radians and the inverse trig functions with A = 32◦ 33 and B = 212◦ 30 , use the − sign; but return angles in radians. The conversion factor is for A = 325◦ 15 and B = 145◦ 17 , use the + sign. π/180 and Excel has a PI() function for the value of π. Perform a linear regression of mλ on both 1. Add a column to your data table converting the of these columns (plus a constant). Then use the Ar and Br for each spectral line to an Hr . Also ﬁtted coeﬃcients to determine d, Hn and Hi . Also convert the incidence angle readings Ai and Bi to record the rms deviation of the ﬁt. an Hi and the grating normal readings An and Bn to an Hn . 3. Make a plot of mλ vs. Hr . Also plot the resid- Recall that the incidence angle θi and the diﬀraction uals: mλ − (Dc sin Hr + Ds cos Hr + D0 ) vs. Hr . angles θr are relative to the grating normal and are thus Misidentiﬁed wavelengths or bad angular measure- given by ments should be obvious from the residuals, which should show only random deviations of less than θ i = Hi − Hn (8) a nanometer centered around zero. Bad points and should be rechecked for possible errors in Hr , m or λ. If you cannot reconcile a bad point, remea- θ r = Hr − Hn (9) sure it. 6 Visible Spectroscopy Advanced Physics Laboratory 3.2. Hydrogen Data 4. COMPREHENSION QUESTIONS Next, you will use the calibration results to determine 2. Compare the ﬁtted value of Hn and Hi with their the wavelengths of hydrogen lines. And then you will use measured values and comment on any discrepancy. these wavelengths to determine the hydrogen Rydberg How close was the ﬁtted d to the manufacturer’s constant. speciﬁcation? Why might they diﬀer? 4. Make a spreadsheet table with a row for each mea- sured hydrogen line having raw data columns for Ar 3. Compare the rms deviation of the ﬁt to expecta- and Br , a column for Hr and one for mλ based on tions based on estimates of the uncertainties in the the regression ﬁt, i.e., using Eq. 11 with the three angular measurements. ﬁt parameters Dc , Ds and D0 from the calibration data. 4. Estimate the rms deviation expected for the ﬁt to 5. Add a column for the order m and use this with the Rydberg formula (assuming, of course, that the column for mλ to determine the wavelength that formula correctly describes the data). Com- associated with the line. ment on the results of the ﬁt to y vs. x with and without a ﬁtted intercept. Discuss the ﬁtted inter- 6. Make any necessary corrections for the index of re- cept and the overall agreement of the ﬁts with the fraction of air, (see the CRC handbook) to obtain Rydberg formula. vacuum wavelengths for the hydrogen lines. 7. Create a second set of columns for determining a 5. Compare your value for the hydrogen Rydberg RH ﬁt of the hydrogen wavelengths to the Rydberg for- with a reference value. Does your result test the mula (Eq. 3). You will need a column for y = 1/λ diﬀerence between RH and R∞ ? Is the correction and one for x = 1/n2 − 1/n2 . Perform a linear f i for the index of refraction of air signiﬁcant at the regression of y on x and plot y vs. x. Equation 3 level of your uncertainties? Explain. predicts that the slope is RH and that the inter- 6. Is the Bohr theory accurate enough for these mea- cept is zero. Thus, while it is instructive to check surements? What is the largest correction to the if the ﬁtted constant comes out zero to within its theory and how does it compare to your experi- uncertainty, the correct statistical procedure is to mental uncertainties? perform the regression ﬁt with the Constant is zero box checked. 7