DECLARE SUB Pager () REM REM GE Copeland REM Dept of Physics REM Old Dominion University REM 2/21/2001 REM 900 REM NATIONAL SCIENCE FOUNDATION 1000 REM COPYRIGHT BY HOUGHTON-MIFFLIN, 1976. ALL RIGHTS RESERVED. 1100 REM "U234" -- (BASIC PROGRAM BEGINS AT LINE 290) 1200 REM 1300 REM 1400 REM DESCRIPTION: 1500 REM 1600 REM THIS PROGRAM ILLUSTRATES THE DOMINANT DECAYS IN THE 1700 REM URANIUM 234 CHAIN. THIS IS BASICALLY A THREE NUCLEII CHAIN. 1800 REM 1900 REM 2000 REM 2100 REM INSTRUCTIONS: 2200 REM 2300 REM THIS PROGRAM IS TAKEN FROM "USING COMPUTERS IN 2400 REM PHYSICS" BY JOHN R. MERRILL AND SHOULD BE USED IN CONJUNCTION 2500 REM WITH THAT TEXT. THE PROGRAMS ARE DISTRIBUTED BY PROJECT 2600 REM CONDUIT. 2605 PRINT CHR$(12); 2700 PRINT "This program is designed to acquaint the student with" 2800 PRINT "radioactive decay schemes. To achieve this let us look at" 2900 PRINT "the DOMINANT decay of the URANIUM 234 chain. But first" 3000 PRINT "let's review general decay concepts." 3100 PRINT 3200 PRINT "The initial unstable nucleus is called the PARENT" 3300 PRINT "and the nucleus into which the parent decays is called the" 3400 PRINT "DAUGHTER. The death of the PARENT gives BIRTH to the" 3500 PRINT "DAUGHTER. The probability that an unstable nucleus" 3600 PRINT "will decay spontaneously into one or more particles" 3700 PRINT "(of lower energy) is independent of the parent's past" 3800 PRINT "history, and is nearly independent of external influences." 3900 PRINT 4000 PRINT "The time at which any one UNSTABLE NUCLEUS will DECAY is" 4100 PRINT "not predicable. But if we treat a large (say a mole)" 4200 CALL Pager 4300 PRINT "number of nuclei, we can statistically determine the" 4400 PRINT "numbers of particles in a decay scheme at any time." 4500 PRINT "During a small time interval dt, the probability that one" 4600 PRINT "will decay is directly proportional to the time interval:" 4700 PRINT "dt=kdt, WHERE k (ARBITRARY SYMBOL) is the decay" 4800 PRINT "CONSTANT i.e. the proportionality constant. Since" 4900 PRINT "the TOTAL PROBABILITY that a NUCLEUS will EITHER SURVIVE" 5000 PRINT "or DECAY in time dt is 100%, we have:" 5100 PRINT 5200 PRINT " PROBABILITY NUCLEUS SURVIVES TIME (dt)=1-k*dt" 5300 PRINT " PROBABILITY NUCLEUS SURVIVES TIME (2dt)=(1-k*dt)(1-k*dt)" 5400 PRINT " =(1-k*dt)^2" 5500 PRINT " THEREFORE, PROBABILITY NUCLEUS SURVIVES TIME (ndt)=" 5600 PRINT " (1-k*dt)^n" 5700 CALL Pager 5800 PRINT 5900 PRINT "Now let ndt=T=Total elapsed time, recalling that" 6000 PRINT "n=T/dt-->INFINITY AS dt-->0," 6100 PRINT "Hence the PROBABILITY that nucleus SURVIVES time " 6200 PRINT "T=LIM (1-k*T/n)^n" 6300 PRINT " n->INFINITY" 6400 PRINT 6500 PRINT "Now by mathematical defination exp(-X)=(1-X/n)^n" 6600 PRINT " n->INFINITY" 6700 PRINT "Thus the PROBABILITY that the nucleus SURVIVES THE TIME," 6800 PRINT "t, is exp(-kt)." 6900 PRINT 7000 PRINT "But if we are dealing with n1 unstable NUCLEI under-" 7100 PRINT "going a decay process characterised by a decay constant k." 7200 PRINT "Then a number n surviving a period t is mearly n1 times" 7300 PRINT "the PROBABILITY that any one nucleus will have survived." 7400 PRINT "this gives n(t)=n1*exp(-kt) (1)" 7500 CALL Pager 7600 PRINT "Usually one measures decays in terms of half life, t(1/2)," 7700 PRINT "which is the time required for one half the original" 7800 PRINT "unstable nuclei to have decayed, we have t=t(1/2) when" 7900 PRINT "n=1/2(n1), and eq. (1) becomes 1/2(n1)=n1*exp(-kt(1/2))," 8000 PRINT "or solving for t(1/2), we find t(1/2)=ln(2)/k" 8100 PRINT "=0.693/k." 8200 PRINT 8300 PRINT "This now ends our discussion on general decay concepts," 8400 PRINT "and we return to the case of U234. This UNSTABLE NUCLEUS" 8500 PRINT "decays in the following manner:" 8600 PRINT 8700 PRINT "234 230 4 226 4 222 4" 8800 PRINT " U---> Th + He--> Ra + He--> Rn + He" 8900 PRINT " 92 90 2 88 2 86 2 " 9000 PRINT " 250KYr 80KYr 1.62KYr <-----half life" 9010 PRINT 9100 PRINT "222Rn then rapidly decayd to 208Pb." 9200 CALL Pager 9300 PRINT 9400 PRINT "In nature U234 is produced by a decay of U238. But since" 9500 PRINT "U238 has such a long half life, we can consider U234 as a" 9600 PRINT "separate CHAIN." 9700 PRINT 9800 PRINT "For single decay systems, or systems in which the half" 9900 PRINT "lifes are very different (vatious members dominate at" 10000 PRINT "different times) an analytical solution can be used:" 10100 PRINT 10200 PRINT " n(t)=n1*exp(-kt)" 10300 PRINT 10400 PRINT "But for other situations the analytical approach is very" 10500 PRINT "complicated (it would involve SEVERAL DIFFERENTIAL" 10600 PRINT "EQUATIONS COUPLED together) to solve; thus we turn to" 10700 CALL Pager 10800 PRINT "numerical methods which are ideal for use with computers." 10900 PRINT "In the U234 case we are using successive Euler approximation " 11000 PRINT "to find our various n's. BUT ENOUGH OF THIS--LET'S GET ON" 11100 PRINT "TO THE PROGRAM. IF ADDITIONAL INFORMATION IS NEEDED SEE" 11200 PRINT "SUBJECT MATTER--NUMERICAL METHODS USING COMPUTERS--IN" 11300 PRINT "LIBRARY." 11400 PRINT 11500 PRINT "At the start of the program you will be given the option" 11600 PRINT "of choosing an initial number of unstable nuclei. The com-" 11700 PRINT "puter will then calculate how the various elements in" 11800 PRINT "the decay chain change with time, i.e. as the unstable " 11900 PRINT "PARENT decays you will see the growth of DAUGHTER" 12000 PRINT "elements). As the tabulated results approach the end there" 12100 PRINT "may be a slight time decay--DO NOT BE IMPATIENT." 12200 CALL Pager 12300 DIM T9(1000), M1(1000), M2(1000), M3(1000) 12400 REM 12500 N8 = 0 12600 N9 = 0 12700 REM * * * * * * * * * * * * * * * * * * * * 12800 PRINT "DO YOU WISH TO PICK AN INITIAL AMOUNT OF UNSTABLE NUCLEI?" 12900 INPUT "(YES or NO)"; a1$ IF LEFT$(a1$, 1) = "Y" OR LEFT$(a1$, 1) = "y" THEN 13200 PRINT "WHAT VALUE DO YOU CHOOSE?"; 13300 PRINT 13400 INPUT I nsave = I 13500 LET n1 = I 13600 IF n1 <= 990000! THEN 13900 13700 IF n1 >= 1E+31 THEN 13900 13800 GOTO 14900 13900 PRINT "THAT WAS A MISTAKE! REMEMBER, THE EQUATIONS" 14000 PRINT "TREAT THE BEHAVIOR OF A LARGE NUMBER OF IDENTICAL" 14100 PRINT "NUCLEI (THIS IS A STATISTICAL APPROACH, AND IT DOESN'T" 14200 PRINT "WORK FOR SMALL NUMBERS). ALSO, WE MUST HAVE A POSITIVE" 14300 PRINT "VALUE! SO ONWARD--CHOOSE A NUMBER BETWEEN 1.E+6 AND .9E+30" 14400 PRINT 14500 GOTO 13200 ELSE END IF 14600 PRINT "I WILL LET N = AMOUNT OF UNSTABLE NUCLEI =1.E^22" 14700 PRINT 14800 LET n1 = 1E+22 nsave = n1 14900 PRINT "DO YOU WANT TO SEE NUMERICAL VALUES (Yes or No)" INPUT c1$ IF LEFT$(c1$, 1) = "Y" OR LEFT$(c1$, 1) = "y" THEN a1 = 1 GOTO 15700 ELSE a1 = 0 END IF 15700 LET N2 = 0 15800 LET N3 = 0 15900 LET T = 0 16000 REM 250 IS HALF LIFE IN THOUSANDS OF YEARS 16100 LET c1 = -.693 / 250 16200 LET C2 = -.693 / 80 16300 LET B2 = -c1 16400 LET C3 = -.693 / 1.62 16500 LET B3 = -C2 16600 LET D = .1 16700 LET T1 = 1 16800 IF a1 = 0 THEN 17500 16900 IF a1 = 1 THEN 17100 17000 PRINT 17100 PRINT "T(KYR)", "N(U234)", "N(TH230)", "N(RA226)" 17200 PRINT "TIME IN" 17300 PRINT "KILO YR" 17400 PRINT 17500 LET L1 = n1 + c1 * n1 * D 17600 LET L2 = N2 + C2 * N2 * D + B2 * n1 * D 17700 LET L3 = N3 + C3 * N3 * D + B3 * N2 * D 17800 LET N3 = N3 + C3 * (L3 + N3) * D / 2 + B3 * (L2 + N2) * D / 2 17900 LET N2 = N2 + C2 * (L2 + N2) * D / 2 + B2 * (L1 + n1) * D / 2 18000 LET n1 = n1 + c1 * (L1 + n1) * D / 2 18100 LET T = T + D 18200 IF T < T1 THEN 19900 18300 REM COUNTER INCREMENT 18400 REM SAVE TIMES AND NUMBERS 18500 N8 = N8 + 1 18600 REM PRINT ONLY EVERY 20 SO WE DON'T BORE THE STUDENT 18700 IF 20 * INT(N8 / 20) = 20 * N8 / 20 THEN 19000 18800 GOTO 19800 18900 REM SKIP TO INCREMENT ON TIME WITHOUT PRINTING OR RECORDING THE DATA 19000 IF a1 = 1 THEN 19200 19100 IF a1 = 0 THEN 19300 19200 PRINT USING " ######.##### "; T; n1 / nsave; N2 / nsave; N3 / nsave 19300 N9 = N9 + 1 19400 T9(N9) = T 19500 M1(N9) = n1 19600 M2(N9) = N2 19700 M3(N9) = N3 19800 T1 = T1 + 1 19900 IF T <= 999 THEN 17500 20000 REM FINISHED CALULATIONS 20100 PRINT 20200 PRINT "Would you like to do a graph of the data (YES or NO)?" 20300 INPUT c$ 20400 IF LEFT$(c$, 1) = "Y" OR LEFT$(c$, 1) = "y" THEN GOTO 21800 ELSE 21500 PRINT "THIS COMPLETES OUR PROGRAM." END IF STOP 21800 REM OPEN OUTPUT FILE FOR TEKTRONIX 21900 OPEN "PLTTEK.DAT" FOR OUTPUT AS #2 22000 PRINT #2, 1 REM!# OF SHEETS 22100 PRINT #2, 3 REM!# OF GRAPHS ON FIRST SHEET 22200 PRINT #2, N9 REM !# OF DATA PAIRS PER GRAPH 22300 PRINT #2, "NUMBER OF NUCLEI 3rd *100 " REM! Y AXIS 22400 PRINT #2, "TIME (KILOYEARS)--->" REM! X AXIS 22500 PRINT #2, " TWO DECAYING-DAUGHTER NUCLEAR CHAIN" 22600 PRINT #2, " 234U->230Th->226Ra" 22700 FOR I = 1 TO N9 22800 PRINT #2, T9(I), M1(I) 22900 NEXT I 23000 FOR I = 1 TO N9 23100 PRINT #2, T9(I), M2(I) 23200 NEXT I 23300 FOR I = 1 TO N9 23400 PRINT #2, T9(I), 100! * M3(I) 23500 NEXT I CLOSE #2 23600 REM FINISHED OUTPUT TO GRAPHICS 23700 PRINT "NOW RETURN TO Windows and run WPLOT." CHAIN "Tek2wplt.bas" 23750 REM 23800 STOP 25500 STOP 26000 STOP 26100 END SUB Pager 25600 PRINT , "Push RETURN to continue"; 25800 INPUT R$ 25806 PRINT CHR$(12); END SUB