The de.i,n and con.truc:tlon of a vacuum high temperature x-ray diffractometer ha. mad. po . . ible tbe inve.tigation of the thermal ex- pan.ion of titanium. Between 00 and 400°C, tb. mean 'coefficient, of thermal expan.ion perpendicular and parallel to the c -axla are
-6,0 -6,0 '
9.41 x 10 C and 11.18 x 10 C, r . . peeUvely. For a random
polycry.talllne .ample, tbe mean Unear expan.ion coeffident 1a 10.0 x
-6,0
010 C. Above 400 C the cia parameter rapidly inc rea ••• a. the tran.formation temperature i. approached, which .upport. a precon- ceived idea about the band Itructure of titanium. Thi. band model ha.
, 0
a . . umed that tbe electronl at tbe top of the band at 0 K are on over- lappin, Fermi .urface. (at lea.t thr •• ) and tbere la a .mall number of unfilled .tate. available to the electron. (approximately 5 x lOl9'c:m3
)
-'"
below an energy lap a . . oeiated witb the c-dlrection in k-.pae •.
Calculation. using thi' model .how that tbe electron. can live an
appreciable po.iUva contribution to tbe frea enar,y a. tb. tran.formation temperature i. approacbed which can r •• ult in an in.tability in the low temperature cry.tal .tructure.
Althou,h tbe analy.ta i. confined to titanium, tbe idea. formulated about allotropic tran.formation. can probably be extended to other
metal • • ucb a. drconlum, hafnium, uranium and iron.
-lZ0-
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-Ill-
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-116-
APPENDIX
WORN aEAR CALIBRA TION
The experimental procedure for worm ,ear calibration requires the use of two worm ,ears • one of which can bo a dummy ,ear. The ,ears rotate about two axes wbicb are almost coincident. A. both ,ear.
rotate 1n tbe .ame direction, the relative anlular motion between the ,oare depends upon the difference of the sum of the cumulative error. in each ,ear. Thb relative rotation can be mea.ured very accurately by tbe chan,tn, frin,e pattern between optical flat. attached to each lear or by a comparison autocollimator and a mirror attached to each ,ear.
The optical flat. technique i. probably a bit too .en.itive, and a relative rotation about an axil otber than the one bein, inve.ti,ated would make the frin,e pattern very complicated. On the other hand, compari.on autocollimatol'l can be obtained which will mea.ure relative rotation about a liven axil witb an accuracy of 0.1 of a .econd of arc.
After mea.urin, the relative rotation of the two ,ea:u, one ,ear i I
rotated with re.pect to tbe other gear a fixed amount, and the experiment repeated. The analy.lI of the experimental re.ult. follow ••
Consider two worm wheel. with n teeth in each lear. A worm i.
enla,ed in each ,ear capable of ,oin, exactly one revolution, which cau.e. the worm lear to rotate an an,le of Z,,/n plus a correction. Let
15
1 be the angular correction to be added to Z.,,/n when aear I rotate.from tooth 1 to tooth 2; 15' Z the correction when aear 1 rotate. from tooth 2 to tooth 3, etc. Tbe notation 5i corruponda to tbe error in Z aear 2 when aear Z rotate. from tooth i to tooth 1+1. When tbe worm
-ll7-
baa ,one through n revolution •• tbe worm ,ear mu.t be back to it.
orilioal po.IUon. Therefore
(A. 1)
After j revolutioD. of the worm, the rotatioD of tb. worm wheel. are
(A. Z)
(A.3)
The quantity that i . mea.ured 18 tbe relative cumuiative error
o'Y/
j' (A.4)or,
(A.5)
Now
(A.6)
U.lo, equation A. 4, equatioo A. 6 become.
J
~ n-l (A. 7)If Gear Z 1. rotated in the direction of deerea.ioa j by "It revolution. of it. worm, while leavina Gear 1 .tationary and the above procedure
-us-
repeated, tbe foUowin, expre •• ion. are obtained:
and
j 0= n-1 (A.8)
Tbere are 2n unknown.. In equation A. 7 and A. 8, j cannot be equal to n, .ince tbi. i. the .ame po.ition a. tbe .tartlns polnt.Tbere- fore, tbere are 2(n~1) Unear equation. from equation. A. 7 and A. 8, and two llnear equation. from equation A .1, giving a total of 20 .imu1taneou.
11n.ar equation. to be .olved. Tbe equation. are rewritten a.
-129-
In tb. above equation. ~jn i. tb. Kroneck.r d.lta. U k i. cbos.n .qual to 1, tbe above •• t of simultaneous linear equation.' can be solved by -deduction. Th. solution. are
n-l
l$j"
o~j
(l-Ojl\) -l'1j-l (l-bj1)-! L
(07[1 -Iii)1 .. 1
(A.9)
Th.r. are various alt.rnativ •• that can be mad. on the experi- m.ntal proc.dure which are . . . . ntially .ubject to the .am. analy.ia.
For in.tanc., ju.t on. mlrror can b. attached to the dummy worm ,ear which h driv.n by a worm attached to the main worm gear aDd rotat.d In a direction oppo.ite to that of the main worm lear. Then the
r
j aremea.ured relative to a .tationary mirror witb a compari.on auto-
collimator. The .ame equation. A.9 apply, but tbe value. of j iner.a.e In oppo.ite direction. on the two ,eare.
Another procedure h to u.e three mirror. and two comparhon autocollimatora a. shown in Figure 43. First, Mirror Z is adju.t.d approximately parallel to Mirror 1. Tb. anll. r.ad on the comparison autocollimator No.1 I.e taken aa its zero position. Tben the main worm lear ia rotated tbroulb an angle of "I ..
!"
+'5
l' Mirror 3 I.e tb.nadjusted approximat.ly parallel to Mirror 1. Tb. anale read on the compari.on autocollimator No. Z i. tak.n a. it. zero po.ition. Mirror.
Z and 3 are not moved a,ain. Th. dummy gear is tben rotated oppo.it.
to the direction of rotation of tbe main worm aear until the &nIle between
Comparison Autocollimator No. I
- 130-
Figure 43.
\
\
(:;?
Mirror 3
Comparison Autocollimator No.2