Statement of Results

Derived wavelength values are shown in the table below. The rightmost column shows absolute error for each derived wavelength, measured against the closest published value.[1] These agree well with probable error, which is based on uncertainty in reading the instrument scale.

Line Wavelengths

Source	Line Description	m	λ, nm	±λ, nm	Δλ, nm
				probable error	compared with published value
H	faint violet	1	410.8	0.2	0.6
	violet	1	434.3	0.2	0.3
	blue-green	1	486.3	0.2	0.1
	red	1	656.3	0.2	0.0
	violet	2	434.7	0.1	~
	blue-green	2	486.7	0.1	~
	red	2	656.5	0.1	~
He	faint violet	1	438.8	0.2	0.0
	violet	1	447.4	0.2	0.2
	faint blue	1	471.7	0.2	0.4
	blue-green	1	501.7	0.2	0.1
	green	1	505.4	0.2	0.6
	yellow	1	587.8	0.2	0.3
	red	1	668.5	0.2	0.6
	faint red	1	707.5	0.2	1.0

Discussion of Results

For mode 2 hydrogen lines, the probable error values came out half the size of those for of mode 1. That seemingly odd result is due to the division by the mode# in calculating wavelength. (See Sample Calculations). The formula for probable error involves the measured wavelength as a factor, causing the result to be divided by m. (A similar outcome multiplies the calculated dispersion for each mode 2 line - see Sample Calculations, and Data Analysis.)

Sodium reference lines at 589.0 nm and 589.6 nm, were used to find the slit separation d (1659.2 ± 0.1 nm). This fundamental value was used in calculating wavelength and probable error in wavelength for non-reference lines, grating lines per cm, grating total lines N, dispersion D, and half-width for the reference lines.

Grating size was found to be 1.53 x 10⁴ lines per inch (6027.0 lines per cm), within 2% of the specified 15,000 lines per inch.

Grating width (2.2 cm) was measured with a ruler, and used to find N, the absolute number of slits in the grating (1.3 x 10⁴). N was then used to compute the half-width angle of each sodium (reference) line, from which was derived a theoretical lower limit of resolution (0.095 nm). An upper or practical limit for resolution is evident in the distance (0.6 nm) between the two sodium lines, which we were able to distinguish with some care.

Spectroscope resolving power R for mode 1, is equal to N (1.3 x 10⁴) by its formula (Nm). R is also expressed as the ratio of wavelength midpoint to wavelength difference, for 2 just-resolvable lines. These were our friendly sodium lines, but R by this method (1 x 10³) was found to be misbehaving by a factor of 10. This was designated as "unhelpful" until it was realized that the wavelength difference used here was 0.6 nm, an upper limit for resolution. Plugging in the theoretical lower limit of 0.095 nm, yields a value for R (6 x 10³ ) that is off only by a factor of 2.

Dispersion D is expressed as the ratio of difference in angle, to difference in wavelength for any two lines. D can also be calcuated for individual lines. Expressed either way, the spectroscope showed dispersion of 60,000 to 70,000 radians per nanometer for mode 1 lines, and 2 to 3 times as much for mode 2 lines.

Scatter plots for Hydrogen and Helium lines (see Graphs) show trends in absolute error. The graph of error vs. wavelength for Hydrogen lines shows decreasing error for increasing wavelengths. The opposite trend is shown on the corresponding Helium graph. This graph appears to show two distinct upward trends - could be random scatter, or could reflect the two of us switching off the task of reading the scale for successive lines.

Error as percentage of wavelength, vs. wavelength (shown for Hydrogen, inspected but not shown for Helium) closely followed the same trends in each case. This suggests that change in wavelength doesn't cause the corresponding change in error.

Significantly, measured wavelengths were almost uniformly greater than published wavelengths. These smooth, distinct trends indicate systematic error, likely associated with instrument alignment, or experimental method, or even the process of assigning values from the reference manual.

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[1] CRC Handbook of Chemistry & Physics, 42^nd Edition (1960)