19_26 - NOMOGRAPH AND TABLE FOR DOPPLER LINEWIDTHS Sidney...

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Unformatted text preview: NOMOGRAPH AND TABLE FOR DOPPLER LINEWIDTHS Sidney 0. Kastner The Doppler width of a spectral line is given by the well—known relation in = (7.162 X 10") AT'” M"”, where wavelength units are in Angstroms, temperature is in degrees Kelvin and M is the atomic mass. This relation between four variables is amenable to representation by a nomograph, which does not appear to have been constructed but which would seem to be of practical value. Therefore such a nomograph is presented here. Its construction is briefly described so that the reader who has not made a plot of this type may follow the steps. The Doppler relation is first rewritten as ltnAh = anA -.~ Val-tn'l' — 95ch or equivalently u, = u, + u; + u,, with k = 7.162 x 10". Puttingé — u. + ilz, first, one has a linear relation 5 - u1 — u: = 0 between three variables which can be represented (Menzel*) by the determinantal equation 5/2 5’2 1 0I l l = 0 u2 fl 1 so that in the Cartesian (x.y) plane the functiOn u2 lies along the x axis, the function u. along the line y: l and the functiOn é/2 along the line y = 1/2. These lines form three parallel scales with which to obtain a value of 5, given the pair (u..uz}. The original equation u. = E + u; provides a second linear relation u.. — 5 — u; = 0, so that a second set of scales similarly results with U3 along the x axis, E along the line y=l and u4/2 along the line y = V2. The function u.. 5 log A). can then be obtained from the known pair (é,u3). In practice, for the nomograph scales to represent useful ranges of the physical variables, some shifting and magnification of the individual scales will be found to be necessary, after completion of the first diagram. The nomograph arrived at here is shown in Figure l. The x axis runs horizontally through the values x=0, T=105, and the ordinates of the six scales from left to right are given by: yI = Sfinflllm kit} y2 = 10Qn(kl\T“’} S y3 = .:. Qn(k1tT“’) y,‘ = Sflnolt + E 531110 y, = Svnt‘r'”;10m) y, = SsntlleM} The approximate range of temperatures covered by the figure is a slowly varying function of wave— length, being 1000°K-10° K for l=l0,000A; 4000 K-5>-<106 K for 1:5000A; 80,000 K-lO? K for i=1000A; and 300,000 K~10’ Kforl=5001~7L In use, the left-hand sides of the three lines are used to find the value of the intermediate variable x corresponding to given A and T. The right-hand sides then give the Doppler width AA in A, for this value of x and the given M. For example, suppose one wishes to find the Doppler width of the solar coronal forbidden line of Fe XIV {atomic mass M =56) at 5303A, emitted at a temperature of about T=2X10‘ K. A straight line drawn between A = 5300A and T=2X105 intersects the x scale at about 4.2. A second line then drawn between it = 4.2 and M = 56 intersects the middle line at Al=0.73A. To use the nomograph, one must know the appropriate atomic weight M for a given element of interest. An alternative and more accurate procedure, if a calculator is available, for obtaining any required Doppler width is to use Table l, which gives the value of the censtant kZ 5 7.162 M5”: for any given atomic number Z, in the equivalent Doppler relation Mt = kztflT'”) X 10". This table thereby avoids the necessity of looking up the atomic weight M. For the forbidden line example above, where Z: 26, the Doppler width obtained from the table is Alt : (0.9584)(5303)(2,000,000)‘“2(10'7} = 0.719A. " D. H. Menze], Fundamental Formulas ofPhysics. Dover Publications, New York, 1960, ch. 3. m a s :- a X 5 5 T M x :3 10’ I w 5 2 2 A 2 1' 10° 3 5 as 03 mm a, 5 " a ' 5 a as 7 0.5 5 a 2 7 0.1 6 3 I U 03 105 1a 12 3 5 14 02 15 IS 2 a 20 -5 0.1 ‘0‘ I am one 4, “m on? 5 5 an a on 1' um 99 a _ 2 m w on: m 5 9o 4 am ‘03 Inn 120 3 5 m am «so 1&1 2 2 m 001 1c2 NOMOGIAPH POI DOPPLER LINEWIDTH II1 UII kfi-hlnd lulu 11.. TI In Find I —l— 1 in ' .l: T in K; M IInIIIi: unifihl [bl Use dam-hand In!” (at. M] an flnd Al TABLE 1 Values of kz (Z Atomic Number) for Use in Doppler Linewidtll Formula A71 = RZOLTM) x 10'T is Z kg 2 kg 2 k; 7.1335 21 1.0682 4| 0.7430 61 0.5927 3.5798 22 1.0348 42 0.7312 62 0.5840 2.7185 23 1.0035 43 0.7201 63 0.5810 2.3857 24 0.9932 44 0.7124 64 0.5711 2.1782 25 0.9663 45 0.7060 65 0.5681 2.0665 26 0.9584 46 0.6943 66 0.5618 1.9137 27 0.9329 47' 0.6896 67 0.5577 1.7905 28 0.9347 43 0.67 55 68 0.5538 1.6431 29 0.8984 49 0.6684 69 0.5510 1.5944 30 0.3853 50 0.65 74 70 0.5492 1.4937 31 0.8577 51 0.6491 71 0.5414 1.4527 32 0.8406 52 0.6340 7'2 0.5361 I .3788 33 0.8274 53 0.6358 7'3 0.5324 I.3514 34 0.8060 54 0.6250 74 0.5282 1.2869 35 0.8012 55 0.62I 2 75 0.5249 1.2649 36 0.7824 56 0.6111 76 0.5193 1.2028 37 0.7747 57 0.6077 77' 0.5166 1.1331 38 0.765! 58 0.6050 73 0.5128 1.1453 39 0.7596 59 0.6033 79 0.5103 1.1313 40 0.7499 60 0.5963 810 0.5057 N s;5:aaz:a=5wmqmuauu— ...
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This note was uploaded on 05/25/2011 for the course CHM A11 taught by Professor Verner during the Fall '10 term at University of Toronto- Toronto.

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19_26 - NOMOGRAPH AND TABLE FOR DOPPLER LINEWIDTHS Sidney...

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