quality corresponding to 13 becomes 
3gkao — 34kKaTo > }eKaW 
— Grae + v4Ko) Ty, (18a) 
It is apparent that the diffusion has 
affected only the last term of the equa- 
tion and all of the mentioned trends 
still are valid. 
Frequency Response 
Returning now to the transfer func- 
tion, it is of interest to obtain order-of- 
magnitude answers for the frequency 
response of the reactor to get a feel for 
the control problem. - For this step one 
must supply numerical values for the 
roots of the characteristic equation. 
As a convenient starting point toward 
obtaining numbers we will first try 
the DD reaction. Post’s calculations 
(Fig. 4) suggest that fission reactors 
operate at a power density of about 100 
watts/em’. At this power density it is 
pointed out that a DD reactor oper- 
ating at 100 kev kinetic temperature 
would need a particle density of 3.3 
10" particles/em’. From these values 
the remaining constants can be 
obtained: 
no?Ke = 6 watts/cm?/kev” 
K, = 1.9 X 107 watt-sec/reaction 
Kp = 1.32 X 10719 watt/kev? 
K,. = 3.45 X 107!9 cm*/sece/kev 
a = 3.05 X 107!” cm/sec 
k = 1.6 X 10716 watt-sec/kev 
Kp is obtained from the steady-state 
situation of Eq. 3 in which it is assumed 
that the diffusion losses just make up 
the difference between the production 
of the thermonuclear charged-particle 
energy and the Bremsstrahlung losses. 
A check of inequality 13 indicates 
that the constants we have chosen rep- 
resent an unstable system since 
3gkaq — 34kKaTo > 14KaW 
a WKg Dy = 
ll 
2K To 
No 
(32 X 10529 — 4:13 XK 10-8 > 65:5 
Le? = 27, XK 105? — 8 <x 1033 
It is evident that at the temperature 
and density chosen the positive tem- 
perature coefficient K, is too large and 
quite dominates the stability picture. 
The depletion-loss term is small com- 
pared with the Bremsstrahlung losses, 
and it appears as though the conditions 
we have chosen do not provide sufficient 
damping to keep the system stable. 
We have the option of changing 7 or 
No in an effort to find better conditions. 
44 
If, as is the case with the DT reaction, 
K, ultimately becomes smaller or even 
negative as the temperature is further 
increased, we might hope to find sta- 
bility at a higher temperature than 100 
kev. Post’s curves give no clue as to 
the magnitude of AK. for substantially 
higher temperatures. Attempts to im- 
prove the stability by lowering no do 
not work within the limits of our as- 
sumptions. That is, with Bohm-type 
diffusion and the assumption that in 
the steady-state diffusion losses make 
up the difference between thermonu- 
clear gain and Bremsstrahlung loss, no 
and Kp vary directly with each other 
and hence a change in density does not 
affect the stability of inequality 13. 
To continue our attempt to assign 
control speeds to the thermonuclear 
reactor, we will now derive the transfer 
function for a DT reactor operating in 
a range where we know the reaction 
has a negative temperature coefficient 
and where it must be stable. In this 
exatople for numerical values we can 
again assume the fission-reactor power 
density of 100 watts/cm*, a tempera- 
ture of 100 kev, and an equal mixture 
of deuterium and tritium with the 
resulting total density of 4.2 x 10! 
particles/cm’. 
The basic stability equations are the 
same as Eqs. 3 and 4, but as shown 
by the table the constants are slightly 
different. The following values were 
derived from Post’s curves: 
Q@ = 8.2 X 10716 em3/sec 
= —5 X 1078 cm*/sec/kev 
5.75 X 10-8 ev 
= 4W = 1.44 X 10-# ev 
Kp =.4.75 X 1078 watt/kev? 
A = —0.531 
B = 4.71 X 1074 
C = —8.09 X 10!2 
D 
E 
rs 
ll 
= —2.15 
F = 2.4 X 10714 
The transfer functions can now be 
obtained directly, and the density func- 
tion becomes 
An(s) _ 
AS(s) 
s+ 5.27 
(s + 24.3)(s + 0.595) 
(15) 
This transfer function is plotted in 
Fig. 3. It is evident that it is similar 
to the transfer function of a fission reac- 
tor. The last break point in the trans- 
fer function occurs at approximately 
4 cycles/sec. In fission-reactor terms 
this would correspond to a reactor hav- 
ing a mean neutron lifetime of about 
3X 104 sec. This equivalent life- 
time might cause the reactor to have a 
speed of response between that of a 
graphite-moderated fission reactor and 
a large water-moderated fission reactor, 
both of which are easily controllable. 
Inasmuch as Fig. 3 shows a finite 
gain at zero frequency the fusion-reac- 
tor transfer function resembles that of 
a fission reactor having a negative 
temperature coefficient. (The trans- 
fer function for a fission réactor with 
zero temperature coefficient has infi- 
nite gain at zero frequency.) Thus all 
of the well-known properties of a fis- 
sion reactor having a slight negative 
temperature coefficient would appear 
to carry over to this type of fusion 
reactor. The reactor in addition to 
being stable would provide a measure 
of self-protection against transients and 
would be incapable of ‘‘running away.” 
Conclusions 
Many other interesting analogies be- 
tween the two types of reactors could 
be derived from the equations pre- 
sented here. It must be remembered, 
however, that the ‘black-box’ ap- 
proach must first be proved. Ques- 
tions as to the effects of changing mag- 
netic fields as well as the possibility of 
having moving boundary conditions 
have been ignored. Our attempt has 
been merely to show one type of engi- 
neering approach to an ultimate con- 
trol problem. 
However, in so far as our assump- 
tions are valid, we have been able to 
establish two things: (1) because of its 
positive temperature coefficient the DD 
reaction will be unstable for tempera- 
tures up to at least 100 kev (about 109 
°K); (2) the DT reaction, on the other 
hand, is stable below this temperature, 
and indications are that the control- 
system speed requirements would not 
be excessive. 
* * * 
The author wishes gratefully to thank J. N. 
Grace for his assistance. 
BIBLIOGRAPHY 
1. R. F. Post, Controlled fusion research—an 
application of the physics of the high tempera- 
ture plasmas, Rev. Mod. Phys. 28, 338 (1956) 
2. M. Kruskal, M. Schwarzchild, Proc. Roy. Soc. 
(London), A 223, 348 (1954) 
3. H. Soodak, E. C. Campbell, ‘‘Elementary Pile 
Theory” (John Wiley & Sons, Inc.,, New 
York, 1950) 
4. M. A. Schultz, ‘‘Control of Nuclear Reactors 
and Power Plants’? (McGraw-Hill Book Co., 
Inc., New York, 1955) 
6. A. Guthrie, R. K. Wakerling, ‘‘The Charac- 
teristics of Electric Discharges in Magnetic 
Fields’’ (McGraw-Hill Book Co., Inc., New 
York, 1949) 
