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4.4.3.1 The Role of Radiation
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4.4.3 Radiation Implosion

4.4.3.1 The Role of Radiation

At the temperatures achievable in the fission core of the primary (up to 10^8 degrees K) nearly all of the energy is present as a thermal radiation field (up to 95%) with average photon energies around 10 KeV (moderately energetic X-rays). Most of this thermal energy is rapidly radiated away from the surface of the "X-ray fireball", composed of the expanding X-ray opaque material of the core and tamper. It is this powerful flux of energy in the form of X-rays that is harnessed to compress the fusion fuel.

To do useful work, the radiant energy from the primary must be kept from escaping from the bomb before the work is completed. This is accomplished by the radiation case - a container made of X-ray opaque (high-Z, or high atomic number) material that encloses both the primary and secondary. The gap between the radiation case and other parts of the bomb (mostly the secondary) is called the radiation channel since thermal radiation travels from to other parts of the bomb through this gap.

The X-ray flux from the primary actually penetrates a short distance into the casing (a few microns) and is absorbed, heating a very thin layer lining the casing to high temperatures and turning it into a plasma. This plasma re-radiates thermal energy, heating other parts of the radiation channel farther from the primary.

The radiant energy emitted by the primary is blackbody radiation: a continuous spectrum of photons whose energy distribution is determined solely by the temperature of the radiating surface. The average photon energy, and the energy of the peak photon intensity are proportional to the temperature. Similarly, the photons re-radiated by the surfaces lining the radiation channel form a blackbody spectrum.

As energy flows down the radiation channel, the energy density drops since the photon gas is now filling a greater volume. This means the temperature of the photon gas, and the average photon energy must drop as well. From an initial average energy of 10 KeV, the X-rays soften to around 1-2 KeV. This corresponds to a temperature in the casing of some 10-25 million degrees K.

[Note: Many descriptions in the open literature exist dating back to the late seventies claiming that energetic X-rays from the primary are absorbed by the radiation casing (or plastic foam), and are re-emitted at a lower energy - implying that some sort of energy down-shifting mechanism (like X- ray fluorescence) is at work. This is a misconception. The lining of the casing is in local thermal equilibrium with the energy flux impinging on it, and re-radiates X-rays with the same spectrum. The X-ray spectrum softens simply because the photon gas cools as it expands to fill the entire radiation channel.]

In physics a closed container of radiation, like the radiation case, is called a "hohlraum". This German word for "cavity" (which has the obvious English cognates "hole" and "room") has been attached to the study of the thermodynamics of radiation since the last century in connection with blackbody radiation. German physicists early in this century used it as a theoretical model for deriving the blackbody radiation laws from quantum mechanics. Energy in a hohlraum necessarily comes into thermal equilibrium and assumes a blackbody spectrum. This is important for obtaining the necessary symmetry for an efficient implosion. Regardless of how uneven the initial energy distribution within the casing is, the radiation field will quickly establish thermal equilibrium throughout the casing - heating all parts to the same temperature.

4.4.3.2 Opacity of Materials in Thermonuclear Design Since the emission, transport, and absorption of thermal radiation is critical to all phases of operation of a thermonuclear device the opacity of various materials to this radiation is critically important. The interaction between a given element and an X-ray photon is dependent on the atomic number, atom density, and ionization state of the element, and the energy of the photon. Since the X-ray flux has a continuous spectrum, we are really interested in the average interaction across that spectrum. We are also especially interested in the situation where the average photon energy (the radiation temperature) and the average kinetic energy of atoms/ions are the same. This situation is called local thermodynamic equilibrium (LTE), and is found almost everywhere inside a thermonuclear device.

The terms "high-Z" and "low-Z" come up frequently in discussing the interaction between thermal radiation and physical materials. These terms are relative - whether a material qualifies as having high or low atomic number depends on the temperature under discussion. The two terms can also be taken as approximate synonyms for "opaque" and "transparent". This is not universally true, however. As explained below, at extremely high pressure this distinction may become unimportant.

An ion strongly interacts with the X-ray spectrum (is opaque to it) when it possesses several electrons, because it then has many possible excitation states, and can absorb and emit photon of many different frequencies. A material where the atomic nuclei are completely stripped of electrons must interact with X-ray photons primarily through the much weaker processes of bremsstrahlung or Thomson scattering. High atomic number atoms hold on to their last few electrons very strongly (the ionization energy of the last electron is proportional to Z^2), resisting both thermal ionization and high pressure dissociation, which is the primary reason they are opaque.

Even when comparing two different fully ionized materials, the higher Z material will more readily absorb photons since bremsstrahlung absorption is proportional to Z^2 (at equal particle density, if it is the ion densities that are the same then it is proportional to Z^3). See Sections 3.2.5 Matter At High Temperatures, and 3.3 Interaction of Radiation and Matter for more discussion of these are related issues.

To summarize: a material qualifies as opaque or "high-Z" if it possesses some electrons at the temperature under consideration. A transparent or "low-Z" material will be completely ionized. Since electrons are removed by ion/particle collisions, the ionization state will depend on the temperature, which is determined by the average kinetic energy (kT) of a particle. At a minimum, all electrons with ionization energies less than or equal kT will be removed, and at the densities of matter encountered here electrons with ionization energies up 3 or 4 kT will often be removed as well.

An important caveat to the above is that at the very high pressures that exist in a fully compressed secondary, essentially any element will become opaque. The density of Fermi degenerate matter under some specific pressure is determined by the density of free electrons. Under the enormous compressive forces generated during secondary implosion the electron density becomes so high that even the "weak" Thomson scattering effect becomes strong enough to render matter opaque. This is important for the energy confinement needed during the thermonuclear burn.

Different temperatures are encountered in different parts of a thermonuclear device - approaching 10 KeV in the primary, 1-2.5 KeV in the radiation channel, and up to 35 KeV inside the secondary. We can make a general guide showing which materials qualify as opaque or transparent at these temperatures by finding Z such that the last (Zth) ionization state has an ionization energy I(Z) approximately equal to kT and 4kT. Any element with I(Z) of around kT will certainly be completely ionized at temperature T. An element will need I(Z) to be significantly greater than 4kT to be highly opaque.

It should be noted that a significant proportion of Planck spectrum energy to be carried by photons with energies even higher than 4kT (10% of it is carried by photons with energies above 6.55 kT). It is possible then at temperatures where the radiation field dominates for the flux of photons to be so intense that photo-ionization of electrons with energies well above 7 kT may occur.

Temperature      Low Z (I(Z)~kT)        High Z (I(Z)~4kT)               Z Symbol  Actual I(Z)   Z Symbol  Actual I(Z) 1 KeV         9    F     1.10 KeV    18   Ar      4.41 KeV 2.5 KeV      13   Al     2.30 KeV    28   Ni     10.7  KeV 10 KeV       28   Ni    10.7  KeV    55   Cs     41.1  KeV 35 KeV       51   Sb    35.4  KeV   101   Md    139.   KeV

From the table we can make some general statements about the materials we want in different parts of the device. We want thermal radiation to escape rapidly from the primary, so it is important to keep the atomic number of materials present in the explosive layer to no higher that Z=28. The use of baratol (containing barium with Z=56) is thus very undesirable. Since the radiation channel needs to be transparent, keeping materials with Z above 9- 13 out of the channel is desirable. Radiation case linings should have Z significantly higher than 55, as should the fusion tamper and radiation shield.

Due to the complexity of the interacting processes that determine the opacity of incompletely ionized material at LTE, theoretical prediction of these properties is extremely difficult. In fact accurate predictions based on first principles is impossible, experimental study is required. It is interesting to note that opacity data for elements with Z > 71 remain classified in the US. This is a clear indication of the materials used in thermonuclear weapon design for containing and directing radiation. The fact that elements with Z > 71 are used as radiation case linings has recently been declassified in the US.

There are 14 plausible elements with atomic number of 72-92 that may be used for this purpose. Of these 14 elements, 5 are definitely known to have been used in radiation case or secondary pusher/tamper designs in actual nuclear devices: tungsten (74), gold (79), lead (82), bismuth (83), and uranium (92). There is evidence that rhenium (75) and thorium (90) may have been used as well, and tantalum (73) has been used in ICF pusher designs. Two others, mercury (80) and thallium (81) are also known to have been incorporated in thermonuclear weapons in classified uses (in addition to declassified uses, such as electrical switches).

The optimal material for radiation confinement should have maximum optical thickness per unit mass. Opacity increases with atomic number, but for a given radiation temperature the increase with Z probably declines at some point. Since atomic mass also increases with Z, there is probably an optimal element for any given radiation temperature that has a maximum opacity per unit mass.

4.4.3.3 The Ablation Process The thin hot plasma layer lining the radiation channel not only radiates heat back into the channel, it also radiates heat deeper into the material lining the channel creating a flow of thermal radiation into the radiation case and the secondary pusher/tamper. The hot plasma also has tremendous internal kinetic pressure and expands into the radiation channel.

This rapid evaporation and expansion (ablation) of the radiation channel lining is unavoidable. Due to the conservation of momentum, the expanding material creates a reaction force called "ablation pressure" that pushes in the opposite direction - blowing the walls of the radiation case outward, and the pusher/tamper of the secondary inward. It is this inward force, analogous to the force exerted by the exhaust of a rocket, that compresses the secondary.

We can calculate representative parameters for the implosion process. To span a range of designs and parameter values let us consider the Mike device, a high yield design that was the first (and undoubtedly physically largest) radiation implosion device ever exploded, and the W-80 cruise missile warhead which is a modern light weight design.

The casing of the Mike device was a steel cylinder 20 ft. (6.1 m) long and 80 in. (2.0 m) wide, with walls 12 in. (30 cm) thick. It used a TX-5 fission primary, with a yield probably no larger than 50 kt, and produced a total yield of 10.4 Mt. The W-80 is a cylinder 80 cm long, and 30 cm wide, it has a primary with a yield in the low kiloton range (call it 5 kt for the sake of the discussion), and a total yield of 150 kt. The thickness of the W-80 casing is unknown, but given its weight (130 kg) it must be less than 2 cm.

Once equilibrium is established, the energy density in the radiation channel will be roughly the energy released by the primary, divided by the volume inside the radiation case (this neglects the kinetic energy in the primary remnants, and the volume of the secondary, but these are comparatively small and offset each other). This gives radiation densities of 2.2 x 10^14 erg/cm^3 for Mike and 4.3 x 10^15 erg/cm^3 for the W-80, a energy density ratio of 1:20. By applying the blackbody radiation laws (see Section 3.1.6 Properties of Blackbody Radiation) we can determine the corresponding radiation intensities and temperatures: 9.8 x 10^6 K and 5.3 x 10^16 W/cm^2 for Mike; and 2 x 10^7 K and 1.0 x 10^18 W/cm^2 for the W-80. The radiation pressures are 73 and 1400 megabars respectively.

The ablation pressure is determined by mass evaporation rate, and the effective exhaust velocity of the evaporated material:
P = m_evap_rate * V_ex
If the evaporation rate is in g/cm^2-sec, and V_ex is in cm/sec, then the result is in dynes/cm^2, applying the conversion factor of 10^6 dynes/cm^2 per bar gives the result in bars.

The ultimate implosion velocity is determined by the rocket equation:
V_imp = V_ex * ln(m_initial/m_final)
where m_initial is the initial mass of the pusher/tamper and m_final is the mass after ablation is complete. Peak efficiency (in terms of energy expended) of an ideal rocket is reached when the ratio (m_initial/m_final) is around 5.

In a rocket maximum force is extracted from hot reaction gases by allowing them to expand as they exit the rocket nozzle, which cools and accelerates the exhaust. The effective exhaust velocity is the velocity of the cooled and expanded gas at the nozzle's mouth. In contrast, ablation is generated by an energy flow that must penetrate the exhaust gas which prevents the gas from cooling. The effective exhaust velocity here is the gas velocity at the sonic point, the point where the gas is moving at the local speed of sound relative to the ablation front, where the material is actually evaporating. Since changes to the exhaust flow beyond the sonic point cannot propagate back to the ablation front, as far as the secondary is concerned the exhaust effectively disappears at this point.

[Note that many descriptions in the open literature ascribe the driving force in implosion to the plasma pressure created by a plastic foam that is known to fill the radiation channel in some weapon designs. Since hydrodynamic effects that occur beyond the sonic point cannot propagate back to the imploding secondary, this is impossible.]

Since the exhaust gases beyond the sonic point absorb heat and carry it away from the secondary, and also reradiate significant amounts of thermal energy back into the radiation channel, the ablation driven acceleration process is less efficient than an ideal rocket as judged in terms of the incident radiation intensity.

The efficiency for an ideal rocket (the percentage of the kinetic energy in the exhaust-rocket system ending up in the rocket at burnout) is given by:
eff = (x (ln x)^2)/(1 - x)
where x if the ratio between the final mass and the initial mass:
x = m_final/m_initial
This has a peak efficiency of 64.8% at x = 0.203.

The heating of the exhaust limits the ablation driven rocket to a maximum efficiency of approximately 15-20% when x is in the range of 0.1 to 0.6 (with peak efficiency around 0.25). Above 0.6 it drops off to about 7% at 0.85. It thus desirable to ablate off most of the pusher/tamper mass so that x < 0.5. [Note: This is based on ICF data which uses radiation driven implosions at a few hundred eV. The higher temperature X-rays of nuclear implosion systems penetrate to the ablation front more efficiently and may actually do better than this.]

Scaling laws for the relationships between temperature or energy density and the ablation rate and exhaust velocity can be determined by dimensional analysis. The sonic-point temperature (and average kinetic energy) is proportional to (~=) the temperature in the radiation channel, and since
v ~= KE^(1/2), 
then 
V_ex ~= T^(1/2).

Because the incident energy flux I between the ablation front and the sonic point must be proportional to the kinetic energy carried away we have:
I ~= m_evap_rate * V_ex^2 ~= m_evap_rate * T
and since
I ~= T^4
we get 
m_evap_rate ~= T^3.

Finally:

P = m_evap_rate * V_ex ~= T^3 * T^(1/2) ~= T^3.5

It is possible to estimate the values of the constants to convert these proportionalities into equations from physical data, but the process is rather elaborate. We can borrow some relationships that have appeared in the inertial confinement fusion literature in connection with radiation implosion to get some estimates of the magnitudes:
P (bars) = 0.3 T^3.5
and
m_evap_rate (g/cm^2-sec) = 0.3 T^3
where T is in electron volts. From this we get:

V_ex (cm/sec) = P/m_evap_rate =  0.3 T^3.5 (10^6 dynes/cm^2 /bar)/0.3 T^3                = 10^6 T^0.5

For the Mike device this gives:
P = 5.3 x 10^9 bars
m_evap_rate = 0.18 g/cm^2-nanosecond
V_ex = 2.9 x 10^7 cm/sec = 290 km/sec

For the W-80:
P = 6.4 x 10^10 bars
m_evap_rate = 1.5 g/cm^2-nanosecond
V_ex = 4.1 x 10^7 cm/sec = 410 km/sec

The ablation pressures for the Mike and W-80 devices are much greater than the corresponding radiation pressures, by factors of 73 and 46 respectively. This shows that the force exerted by radiation pressure is comparatively small.

From the classical rocket equation given above we can estimate V_imp at maximum efficiency (where 75% of the mass is ablated off) at 400 km/sec (Mike) and 570 km/sec (W-80).


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