Limits of precision in the Balmer lines spectroscopy lab

Balmer lines spectroscopy can reach an impressive level of precision even with the common student spectrometer. Understanding the sources of error can help the instructor in guiding and assessing student work as well as in teaching about error analysis itself. We analyze the most significant contributions to error, both random and systematic, including approaches to minimize or measure them. This can be applied to teaching labs whether accuracy is ~ 1% (introductory level) or better than 0.05% (in-depth advanced level).


I. INTRODUCTION
In this paper we examine the most significant sources of error that arise in measurement of the Balmer lines with a diffraction grating spectrometer, with a focus on applications to teaching a student laboratory course.An instructor's understanding of the causes and sizes of errors (both systematic and random) helps inform the experimental and analytical approaches for the lab, which can lead to better student understanding of their data.
We present our analysis of errors (or uncertainties, if you prefer) in the context of an example experiment, taken from a two-week investigation in our sophomore-level Modern Physics Laboratory course.We give what we hope is sufficient derivation so that our results can be easily adapted to other experimental approaches with similar equipment.
During the first week of the experiment, we review the basics of diffraction and the students learn how to measure the diffraction angles using a grating spectrometer [1].They calibrate the instrument by measuring angles for light from a Hg discharge lamp, which has lines of known wavelength [2].Their analysis gives a value for the grating constant d and uncertainty d ∆ .In week two, they measure angles of the Balmer lines from a hydrogen lamp, from which they determine the wavelengths.
These are compared qualitatively to the Bohr model and quantitatively to either the ionization energy or almost equivalently, to the Rydberg constant.
For a short intermediate or introductory lab, an accuracy of 1% may suffice and our analysis shows where the most significant errors might arise.However for a more in-depth lab like ours, errors can be brought down to ∼ 0.03% even using a student spectrometer and physical effects such as the reduced mass and index of refraction of air become significant and worthy of discussion and correction.

II. DIFFRACTION THEORY
A common type of spectrometer for the student lab uses a transmission diffraction grating, with light ideally arriving normal to the grating and beams due to constructive interference emerging as shown in Fig. 1.The light source is typically a gas discharge lamp from which some light is passed through a narrow slit aperture and then formed into a parallel beam by a collimation lens.In the normal-incidence situation, the diffraction angles are given as where d is the diffraction grating constant (= inverse of the # lines/meter).

FIG. 1 Diffraction of monochromatic light into different orders (grating aligned for normal incidence).
A value for the wavelength can be determined from a measured diffraction angle as ( ) where i θ is the angle of the incident beam relative to normal.
Fig. 2 shows the situation for non-normal incidence, for outgoing beams 0, 1 m = ± .Note that when i 0 θ > as shown, the 0 m = beam is at a negative angle with respect to the normal, but equal in magnitude to i θ .
Experimentally, it is difficult to determine the desired beam angle with respect to normal, m θ , in this situation.Many spectrometers give an angle reading ϕ with respect to an angle origin that is never perfectly aligned with the grating or the incident light direction, but we can make use of the Usually one tries to arrange that i θ is so small that we can ignore its effect on calculation of the wavelength.In Section B we look at the size of the error introduced and one approach for minimizing the error.

A. Normal Incidence
Let's first look at Eq. (1.1), applicable if the grating is well-aligned to be normal to the incident light.A first order expansion considering small uncertainties , (1.4) The first term represents the relative uncertainty in the grating constant while the second term represents the relative uncertainty in the angle measurement.We will consider these two contributions separately.

Error from angle measurement
The uncertainty or error arising from measuring the diffraction angle we write as Eq.(1.5) shows that the relative uncertainty in wavelength is proportional to cot m θ , so the larger the angle the smaller the error in wavelength.However, students may not have much familiarity with cotangent and so lack instinct for the trend unless time is taken to examine it.
Eq. (1.6) shows that for a given wavelength, the absolute uncertainty in λ decreases with increasing m .Thus there is a strong advantage to using higher orders, if they exist within the range of diffraction, restricted by / 1 m d λ < .The uncertainty decreases even more with angle as the cosine term decreases, but this is not so significant unless you go to angles larger than 45 degrees.
An example of the error propagation results is given in Table I, for light near the middle of the visible spectrum (specifically, wavelength 546nm, the green mercury line) and for a grating with 300 lines/mm ( 3333 d  nm).It assumes a measurement error m θ ∆ = 1 minute of arc = 0.29mrad, the smallest measurement division on many student spectrometers.For a student with just a little practice, the typical angle measurement error is at least two times larger.

Error from grating constant (calibration error)
The grating constant might be obtained from the manufacturer or as we do, by calibration using a known light source.The first term in Eq. (1.4) shows that uncertainty in the grating constant goes directly into uncertainty in wavelength, This is usually a systematic error, since one value of grating constant is used throughout an experiment.

B. Non-normal incidence: Grating Misalignment
We suppose there is an unknown residual misalignment i θ and we make a measurement of the difference between the m th order and 0 th order beams, (1.7) We might use m φ to calculate wavelength assuming no misalignment, thence obtaining a value ( ) How large is the error caused by i θ ?
We begin by re-writing (1.7) to obtain ( )  where for convenience of notation we introduce the quantity With these, our calculated wavelength becomes ( ) ( ) Eq. (1.8) can be expanded as a Taylor series in i θ , a lengthy process which gives, to 2 nd order, ( This has a first-order error in the misalignment angle, with coefficient ( ) ( ) Thus for a given misalignment the relative error increases roughly linearly with order and wavelength.An example of the wavelength error for several orders as a function of misalignment is given in Fig. 3.If there is a discernable (significant) grating misalignment then conceivably one could measure itessentially calibrate it -using a light source of known wavelengths and the model of Eq. (1.7).We use instead a simple and intuitive method to eliminate the first-order error, by combining two orders m a = ± .We then compute  .Combining this with Eq. (1.7) and once again expanding as a series to 2 nd order in i θ gives ( ) .11) which is only 2 nd order in misalignment.Interestingly but perhaps not surprisingly, the quadratic coefficient ( ) is numerically very nearly equal to the quadratic coefficient in (1.9), the one-sided expansion: ( ) and increase only slowly in this range.A comparison of the errors arising from this computation method is shown in Fig. 3, where it is clear that it gives much smaller error due to misalignment than a onesided measurement.The error for 4 m = becomes greater than the best possible precision error (Table 1, 0.18nm or 0.03%) only for misalignments greater than about 0.02 rad.

IV. OTHER PHYSICAL EFFECTS
The measurements described give wavelengths in air.If students compare their results to other measurements (for example, Balmer wavelengths from NIST), they should be sure to check if the values are for air or vacuum.If quantities relating to the structure of hydrogen are to be compared (for example, calculating the Rydberg constant from measured Balmer wavelengths) then the wavelengths should be converted to vacuum values first.These are related by FIG.2Angle definitions for case of a misaligned grating.

FIG. 3
FIG. 3 Wavelength error from grating misalignment.Colored lines are measurement using single order (blue m a = + , red m a = − ).Black lines are for combined orders & m a a = + − , resulting in smaller errors.Values are for wavelength 546nm, d = 3333nm.
of similar size is that of the hydrogen nuclear mass, accounted for with the theoretical model of reduced mass.The Rydberg constant most often quoted

Table I :
Wavelength error for different orders, caused by an error in measurement of diffraction angle of 1' (  0.02° or 0.3mrad).