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BREMMSTRAHLUNG RADIATION LOSSES IN POLYWELL(tm) SYSTEMS

EMC2-0891-04

Robert W. Bussard and Katherine E. King

__________________________________
This work performed under Contract No. MDA-972-90-C-0006 for the Defense Advanced Research Projects Agency, Defense Sciences Office.

92-29999 [handwritten annotation: 8PJ]

EMC2(r)ENERGY/MATTER CONVERSION CORPORATION
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Page 1 wrote:BREMMSTRAHLUNG RADIATION LOSSES IN POLYWELL(tm) SYSTEMS

The degree to which bremmstrahlung radiation constrains or limits system performance depends on the energy E_e of the electrons which are the principal source of this radiation, through their collisions with in-situ ions. This is true in those cases in which the electron energy is such that the electron speed exceeds the ion speed, at which condition the ions can be regarded as stationary targets for the electrons. If the electron energy is so low that the electron speed is comparable to the ion speed, then the ion energy must also be taken into account in computation of bremmstrahlung.

This latter condition can obtain only in the core of the Polywell(tm) device, and then only if the central virtual anode is nearly zero in height, and if the electron/ion collision rates are sufficiently small that no significant collisional heating of electrons can take place during the electron lifetime in the machine. Conditions under which these effects can be achieved in the Polywell(tm) system (in marked contrast to magnetic confinement LTE systems; in which they can NOT be achieved) are discussed further, following.

The basic expression for bremmstrahlung power density(ref.1) in a mixture of electrons and j classes of ions, each with charge Z_j,

q_br = 1.69E-32 (E_e)^0.5 n_e Sum(n_j Z_j^2) eq.(1)

shows this quite clearly. Here E_e is in eV and n in 1/cm^3 for q_br in w/cm^3.

(ref.1) D. L. Book, "NRL Plasma Formulary," 1983 rev., Naval Research Laboratory, Wash., D.C., p. 57

In considering the range of fusion fuels it is important to note that the effect of Z > 1 can become quite profound on bremmstrahlung output, event at small Z. This is because, in the Polywell(tm) system, the charge density is very nearly neutral in the regions of highest density, where the principal radiation is generally, and n_e = n_1 Z_1 + n_2 Z_2 while Sum(n_j Z_j^2) = (n_1 Z_1^2 + n_2 Z_2^2), where n_1 and n_2 are the local densities of the two fusion fuel species.

Writing these as fractions of the total ion density n_i, n_1 = f_1 n_i, n_2 = f_2 n_i, gives the bremmstrahlung power density as

q_br = 1.69E-32 (E_e)^0.5 n_i^2 (f_1 Z_1 + f_2 Z_2) (f_1 Z_1^2 + f_2 Z_2^2) eq.(2)

Now, the total bremmstrahlung power output is just this expression integrated over the total ion density and electron density distribution in the system volume. Thus P_br = Integral( q_br 4 pi r (exponent of r illegible) dr) over 0 <= r <= R. It is readily shown(ref.2) that this can be written in a simpler form as the integration over the convergence core volume, multiplied by a factor K_b which is the ratio of total bremmstrahlung power to core region power, thus K_b = P_br(total) / P_br(core)

(ref.2) R. W. Bussard, G. P. Jellison, G. E. McLellan, "Preliminary Research Studies of a New Method for Control of Charged Particle Interactions," Pacific-Sierra Research Corp., Report PSR 1899, 30 November 1988, Final Report under Contract DNA001-87-C-0052

Page 2 wrote:Assuming that the fuel mixture remains constant(fn1) throughout the region of the machine that is effective for generation of bremmstrahlung, this factor can be found by integration of eq.(2) with the appropriate distributions. In the region 0 <= r <= r_c, both the ion density and the electron energy may be taken as constant, thus n_i(r) = n_c, and E_e(r) = nu_e E_o, where E_o is the electron injection energy (and maximum possible well depth). The parameter nu_e = E_e(core) / E_o is the fractional energy of the core electrons, expressed as a virtual anode height parameter. Note that nu_e > nu, where nu is the height of the central virtual anode.(ref.3)

(fn1) The fuel mixture will NOT be uniform over the total volume in mixtures of high-Z and low-Z fuels; high-Z fuels will be excluded from the outer regions of the machine. However, essentially all of the bremmstrahlung comes from the near core region, where the fuel mixtures will be constant.

(ref.3) R. W. Bussard and K. E. King, "Electron Transit Time in Central Virtual Anode Wells," Energy/Matter Conversion Corp. Technical Note, EMC2-0291-03, Mar. 1991

From r_c < r < r_k the density follows n_i(r) = n_c ( r_c/r )^2 and the electron energy can be taken (see ref.3) as varying as E_e(r) = nu_e E_o ( r_c/r )^2 + E_o ( r/R )^3, for a potential well with m = 3.(fn2) This gives a reasonably good fit to the local potential and thus to the local electron energy, which is assumed to be in equilibrium with the potential. It can be shown(ref.4) that the ion density increases in the region from r_k < r < R from its value of n_k at r_k to n_R = 3.0 n_k. For convenience this can be written as n_i(r) = n_k ( r/r_k )^q, where the exponent is given by q = 3.0 / [LN(R/r_k)].

(fn2) The actual well in an m = 3 system follows the "rollover" formula <r>^3 f_o(r), where f_o(r) = 2 / (1 + <r>^5). However, most of the bremmstrahlung comes from the inner regions where r << 1, in which the <r>^3 approximation is quite good.

(ref.4) R. W. Bussard, "Edge Region Distributions and Synchrotron Radiation," Energy/Matter Conversion Corp. Technical Note, EMC2-0991-04, Sept. 1991

Using these forms and integrating it is found that the first and second terms (i.e. in the region r < r_k) are dominant, and that the bremmstrahlung is split about 40% from the core and 60% from the central region immediately outside the core within the intermediate and inner mantle region, r << r_k. This result agrees with previous analyses(ref.5) of the distribution. Thus K_b = 2.50 and the bremmstrahlung power is given by

P_br = 1.69E-32 [ (f_1 Z_1 + f_2 Z_2) (f_1 Z_1^2 + f_2 Z_2^2) ] [ K_b n_c^2 (E_e)^0.5 ] eq.(3)

in watts for E_e in eV, where n_c is the ion (and electron) density in the core.

(ref.5) Op cit ref. 2, App. A, Sect. 2(d)

The total fusion power in the system can be written in terms of the local fusion power density

q_f = b_ij(n_i(r))^2 sigma_f(E) v_i(E) E_f eq.(4)

integrated over the system. Here b_ij accounts for the possible number of interactions among differing and like specie fuels. In like fuels (e.g. DD) b_ij = 0.5, and for unlike fuels (e.g. DT) b_ij = f_1 f_2 = f_1 (1-f_1) = f_2 (1-f_2). The maximum value of b_ij for unlike fuels requires that f_1 = f_2 = 0.5 for which b_ij = 0.25.

In a similar fashion to the bremmstrahlung analysis, above, the total fusion power can be expressed in tgerms(sic) of that generasted(sic) within the core and that outside, by the ratio K_f = P_f(total)/P_f(core). However, here the ion energy distribution differs from that for the electrons, as the system is nowhere in LTE and the ions are "cold" where the electrons are "hot", and vice versa. The ion energy varies as E_i(r) = (1-nu) E_o ( r_c/r )^2 + E_o ( r/R )^3 = E_o (1 - <r>^3) - nu E_o ( r_c/r )^2 .

Page 3 wrote:Detailed calculations of fusion power density distribution and total fusion power output have been made for a variety of systems, using the EKXL v.4.1 code. Results(ref.6, ref.7) of these show that both the power density distribution and total power are functions of the central virtual anode height (and thus of the allowed ion current). The variation is such that, as the anode height increases, less of the fusion power is generated within the core convergence radius <r_c>, and more comes from the region immediately outside (r < 10 r_c) this core. As the anode height factor (nu) approaches unity, the core-generated power drops to zero and all of the power comes from outside the core. From this work it is found that the factor K_f varies as

K_f = 2 (1 - nu_o) / (1 - nu) eq.(5)

where nu_o is that value at which the in-core and out-of-core contributions are equal. Typically, nu_o ≈ 0.167.

(ref.6) Robert W. Bussard and Katherine E. King, "Virtual Anode Height Variation in Poywell(tm) Systems," Energy/Matter Conversion Corporation Technical Note, EMC2-0991-02

(ref.7) EMC2 Presentn(sic) to Meeting of HEPS Program Sci. Advisory Board, San Diego, CA, 29, 30 July 1991

With this and noting the ion collisional speed in the CM system as given by v_i = (2 E_i / m_pi)^0.5, where m_p is proton mass and M_i = m_1 m_2 / [ (m_1 + m_2) m_p ] is the normalized reduced mass of the ions, the total fusion power can be written as

P_f = 0.1 b_ij K_f n_c^2 [ sigma_f(E_c) ] [2 E_c / (m_p M_i) ]^0.5 E_f k_e^1.5 eq.(6)

in watts, for k_e = 1.6E-12 ergs/eV, the fusion reaction energy E_f in MeV, and the core ion energy E_c in eV. Net power output requires that the ratio P_fb = P_f / P_b be greater than unity. From eqs.(4) and (6) this becomes

P_fb = eq.(7)
K_f b_ij [ 2 / (m_p M_i) ]^0.5 (sigma_f E_f) k_e^1.5 (E_c)^0.5
-------------------------------------------------------------------------
K_b [ F_2(Z) ] 1.69E-31 (E_e)^0.5

where F_2(Z) = (f_1 Z_1 + f_2 Z_2) (f_1 Z_1^2 + f_2 Z_2^2).

Specializing to the case where one fuel is singly-charged (Z_1 = 1) and noting that f_1 + f_2 = 1, gives F_2(Z) = [ 1 + (Z_2-1)f_2 ] [ 1 + f_2 (Z_2^2 -1) ] . With this and writing E_c = (1 - nu) E_o, E_e = n_e E_o eq.(7) becomes

P_fb = eq.(8)
K_f [ F_3(Z) ] [ 2 / (m_p M_i) ]^0.5 (sigma_f E_f) k_e^1.5
--------------------------------------------------------------------
K_b 1.69E-31 [ nu_e / (1-nu) ]^0.5

Here the function F_3(Z) = b_ij / F_2(Z) = (1-f_2) f_2 / F_2(Z). Evidently there is an optimum value of the high-Z fuel fraction f_2, that will give a maximum fusion-to-bremmstrahlung ratio. This is found by differentiation of F_3(Z) to be

f_2|opt = 1/( Z_2^1.5 + 1 ) eq.(9)

and the optimum ratio of Z_1 = 1 to high-Z fuels is f_12 = f_1 / f_2 = Z_2^1.5.

Thus, for D^3 He, f_2 = f_He = 0.261, f_12 = 2.83, while for p^11 B, f_2 = f_B = 0.082, and f_12 = 11.2. In the D^3 He case the system must be rich in D, which leads to [a] larger fraction of DD reactions and thus to higher neutron radiation output than for 50:50 or lesser mixtures. The p^11 B case is very proton-rich, which leads to much smaller power output from a given size of device which will, in turn, drive the system to larger sizes and higher B fields.

Page 4 wrote:The maximum value of P_fb is thus determined by the natural properties of the fuels, the fusion cross-section (thus by the injection energy and well depth) and the energy of electrons in the central core. Using eqs.(5) and (8) and taking operation at optimum conditions (eq.9), the ratio P_fb can be written as

P_fb = [ F_b(f, Z, M_i, E_f) ] [ sigma_fb(E_c) ] [ 2 (1-nu_o)/K_b ] [ 1 / nu (1-nu) ]^0.5 eq.(10)

for sigma_fb in barns (b), taken at core ion energy, E_c = (1-nu) E_o, and the core electron energy factor has been set at nu_e = nu. The inherent values for the functional term, F_b are given in Table 1, below, for optimum mixtures and for 50:50: (equals) mixtures of each of the fuels shown. Also shown is a very He-rich D^3 He case, to approximate "radiation-free" (i.e. insignificant DD reactions) operation such that NO shielding is required with this fuel combination.

Code: Select all

TABLE 1
FUSION-TO-BREMMSTRAHLUNG FACTORS FOR VARIOUS FUELS

                Optimum Fuel Mixtures         50:50 Mix         1:1000
Fuel        DT    DD        D^3 He  p^11 B    D^3 He  p^11 B    D^3 He
------------------------------------------------------------------------
E_f (MeV)   17.6  3.65      18.3    8.7       18.3    8.7       18.3
M_i         1.20  1.00      1.20    0.92      1.20    0.92      1.20
f_2|opt     0.50  --(note2) 0.26    0.082     0.5     0.5       0.999
------------------------------------------------------------------------
F_b (note1) 57.7  23.9      18.8    2.28      13.0    0.76      0.22
------------------------------------------------------------------------
(note1) for sigma_fb in (b)
(note2) b_ij factor for DD is 0.5

Note that the energy per fusion event is lower for DD than is frequently quoted(ref.8, ref.9) for complete burning of all the products of the initial DD reaction. This is because the fusion products always escape the core of the electrostatic system and are not used directly in the burn cycle within the confined core region. Also note that the F factor for D^3 He drops drastically as the mixture ratio is changed to seek nearly-neutron-free fusion power gfeneration(sic), so that D^3 He systems than(sic, that) can be operated without significant radiation shielding have F values less than those for p^11 B, which has no direct neutron output.

(ref.8) Robert A. Gross, "Nuclear Fusion," John Wiley & Sons, New York, 1984, Chaps. 1-3

(ref.9) S. Glasstone and R. Lovberg, "Controlled Thermonuclear Reactions," Van Nostrand, New York, 1960, Chaps. 1, 2

Since K_b = 2.5, n_o = 0.167, and nu must be small for effective operation, eq.(10) can be approximated as

P_fb= 0.667 F_b sigma_fb / nu^0.5 eq.(11)

For fusion power generation to exceed bremmstrahlung then requires that P_fb> 1, which will occur only when sigma_fb > K_b (nu (1-nu))^0.5 / [ 2 F_b (1 - nu_o) ] = 1.34 nu^0.5 / F_b, as a necessary criterion for net fusion power.

Page 5 wrote:Determination of the minimum possible value of nu_e that can be achieved is of some complexity, for it involves consideration of ion/electron up- and down-scattering collisions in different regions of the system, as well as of the ratio of ion current to electron drive current used to establish and maintain the potential well.

The EXKL code runs have shown that minimum virtual anode heights will always be at or above (1-alpha_q) ≈ 0.005 (i.e. the maximum well depth never gets closer to injection energy than alpha_q = e theta_max / E_o ≈ 0.995). If the electrons could be kept at this energy, then nu_e = 0.005, and bremmstrahlung losses will always be much less than fusion power generation capabilities The question here is the degree to which ion/electron collisions in the core region can transfer ion energy to the electrons sufficient to raise nu_e significantly above this level. It is thus necessary to examine the ion/electron collisional energy exchange process in some detail.

Such energy exchange, will, of course, occur in the core, mantle and edge regions. In the outer mantle, beyond the electron "stagnation" radius <r_f> = (dE_o1 / E_o)^0.6 and in the edge region, electron/ion collisions will "cool" the electrons, while in the region inside <r_f> ions will "heat" electrons. The important feature is the balance between up- and down-scattering in a single pass of an electron through the system. If the up-scattering in core region passage is removed by the down-scattering in extra-<r_f> collisions with cold ions, then the core electron energy will be stable. This stable electron energy is thus a result of competing collisional processes in the spatially-alternating non-LTE ion/electron distribution of the system. Analysis of these processes shows the stable up-scattered electron core energy (at which equality of ion/electron energy exchange will take place in the "heating" and "cooling" sections of the system) to be approximately

nu_e = (N_core / N_tot)( <r_f>^4 ) / ( 10 <r_c>^0.5 ) eq.(11) [ed. note: there are two eq.(11)]

Here N_core / N_tot is the fraction of electrons that "see" the core, equivalent to the single-pass core-sampling frequency of electrons circulating in the system, and the factor of 10x comes from analysis of the edge/mantle cooling collisions. The sampling frequency can be estimated by a simple ratio of electron number in each region. Using the density distributions cited above and integrating over the complete system gives this approximately as <r_c>/3. Substituting into eq.(11) yields

nu_e ≈ ( <r_f>^4 ) <r_c>^0.5 / 30 eq.(12)

If <r_f> = 0.707, for example (a highly-spread electron distribution), and the convergence radius is take to be <r_c> = 1E-2, then nu_e= 0.83E-3 is found. If <r_f>= 1.0 (the maximum possible value), then nu_e = 3.3E-3, above the ion-driven virtual anode height. Taking this as nu = 5E-3, as discussed above (for maximum alpha_q), yields an electron core energy of nu_e = 5.8[E-3] - 8.3E-3 as an absolute minimum for a system constrained to operate at the lowest possible virtual anode height.

Page 6 wrote:It is obvious, from eqs.(10,11), that maximum P_fb will be found for the highest possible value of the fusion cross-section, sigma_fb, thus for operation at that energy at the peak of cross-section variation. However, this peak energy is not necessarily optimum for the competition of fusion with synchrotron radiation power, P_sy. A study of the synchrotron question has shown(ref.10) that the optimum well depths (injection energies) for maximum P_fs = P_f / P_sy are as listed in Table 2, below.

(ref.10) Op cit ref. 4

Fusion-to-bremmstrahlung power ratios, P_fb, are given in Table 2 for each of the fuel mixtures used in Table 1, above, for a variety of well depth or injection energy conditions. These calculations have been made using a practical minimum value of nu_e = 0.01 for the core electron energy ratio; from eq.(11) this gives P_fb = 6.667 F_b sigma_fb .

Code: Select all

TABLE 2
OPTIMUM OPERATION FOR FUSION/BREMMSTRAHLUNG POWER BALANCE

                        Optimum Fuel Mixtures         50:50 Mix         1:1000
Fuel                DT    DD        D^3 He  p^11 B    D^3 He  p^11 B    D^3 He
--------------------------------------------------------------------------------
E_f (MeV)           17.6  3.65      18.3    8.7       18.3    8.7       18.3
f_2|opt             0.50  --(note2) 0.26    0.082     0.5     0.5       0.999
F_b (note1)         57.7  23.9      18.8    2.28      13.0    0.76      0.22
--------------------------------------------------------------------------------
for opt synchrotron losses
E_syn (keV)         30    23        110     500       110     500       110
sigma_fb(E_syn) (b) 4.0   0.030     0.50    0.70      0.50    0.70      0.50
P_fb|syn            1539  4.78      62.7    10.64     43.3    3.55      0.73

for optimum bremmstrahlung
E_pk (keV)          40    600       170     560       170     560       170
sigma_fb(E_pk) (b)  5.0   0.20      0.70    0.80      0.70    0.80      0.70
P_fb|max            1923  31.86     87.7    12.16     60.7    4.05      1.03
--------------------------------------------------------------------------------
(note1) for sigma_fb in (b)
(note2) b_ij factor for DD is 0.5

Note from Table 2 that all of the fuels can operate at bremmstrahlung-optimum mixture ratios with negligible bremmstrahlung losses if the electron energy state can be kept as low as assumed above. However, losses with p^11 B at 50:50 mixtures are significant in comparison with fusion power generation, and losses in D^3 He at the 1:1000 mixture ratio taken for radiation-free operation are prohibitive. DT is able to operate easily at all conditions, and can function quite well at any virtual anode height condition. In fact, all of the fuels at optimum mixture conditions can operate with minimal bremmstrahlung at anode heights of nu_e <= 0.15, or so. Also, it is clear that DD and D^3 He offer similar performance envelopes at optimum mixture conditions while D^3 He is similar to p^11 B when operated in a non-radiative mode.

Page 7 wrote:Finally, note that bremmstrahlung is a more pervasive constraint than synchrotron radiation because the latter can be reflected by bounding metal walls, and the loss fractions are much less than given above when the effects of resonance self-absorption within the plasma are taken into account.(ref.10) Bremmstrahlung power generation is inherent in the plasma mixture, it can not be suppressed, reflected or self-absorbed - it is simply a loss mechanism.

In conclusion it is gratifying to see that all four of the fuel combinations can be made to work effectively in the Polywell(tm) system; a result that is not true for use of these fuel combinations in "conventional" magnetic, Maxwellian fusion systems in local thermodynamic equilibrium.

page 1 wrote:P_br = Integral( q_br 4 pi r (exponent of r illegible) dr) over 0 <= r <= R

dch24 wrote:I don't know why this is in News; Art, maybe you can clarify that? But the scanned document (EMC2-0891-04) could use re-typesetting, so here goes:

Art Carlson wrote:If there is a closed cycle of energy transfer like this:
core ions -> core electrons -> halo electrons -> halo ions -> core ions ...
then you could build a perpetual motion machine by tapping into those energy flows. You could extract high-quality (low-entropy) energy and replace it with low-quality heat.

The suppression of bremsstrahlung is real, but the cost is high, in the sense defined above that the recirculating power for a non-Maxwellian distribution is prohibitively high. If there is some clever way of recirculating that power with extremely small losses, it hasn't been published yet. I can't imagine any. Neither could Rider, although he tried a lot harder than I did. A serious defense of a p-B11 polywell has to address this issue.

• at what rate must electrons be lost and re-injected,
• how are they lost (among other things, how is it ensured that electrons with the proper energy are lost, and only those),
• how are those electrons and their energy collected and with what efficiency,
• and how are they re-injected and with what efficiency.

93143 wrote:

The suppression of bremsstrahlung is real, but the cost is high, in the sense defined above that the recirculating power for a non-Maxwellian distribution is prohibitively high. If there is some clever way of recirculating that power with extremely small losses, it hasn't been published yet. I can't imagine any. Neither could Rider, although he tried a lot harder than I did. A serious defense of a p-B11 polywell has to address this issue.

...so you don't consider the posited "annealing" effect to be worthy of notice as a possibility? Remember, the Second Law only applies to the whole system, not to a single pass through any individual region.

Art Carlson wrote:Annealing never made much sense to me, but even if it works as advertised, I thought it was supposed to be a way to keep the ions focussed. I have never heard that it is also supposed to keep the hole in the middle of the electron distribution clear.

93143 wrote:The electrons are fast at the edge, and fairly slow in the core. They are slowest at the bottom of the well, which is not at the centre of the reactor but rather at some finite radius, where the ion focus begins to dominate the potential distribution and the virtual cathode starts to give way to the virtual anode. The electron density should also have a maximum very near this radius, because of the 'traffic-jam' effect of the slowdown.

The result is a region with low energy, high density, and high cross section, just like the edge region for the ions.