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88-540: The TFTR never achieved this goal, but it did produce major advances in confinement time and energy density. It was the world's first magnetic fusion device to perform extensive scientific experiments with plasmas composed of 50/50 deuterium/tritium (D-T), the fuel mix required for practical fusion power production, and also the first to produce more than 10 MW of fusion power. It set several records for power output, maximum temperature, and fusion triple product . TFTR shut down in 1997 after fifteen years of operation. PPPL used

176-416: A fusion triple product of 1.5 x 10 Kelvin seconds per cubic centimeter, which is close to the goal for a practical reactor and five to seven times what is needed for breakeven. However, this occurred at a temperature that was far below what would be required. In July 1986, TFTR achieved a plasma temperature of 200 million kelvin (200 MK), at that time the highest ever reached in a laboratory. The temperature

264-453: A commercial system followed, that could be built at Oak Ridge. They gave the project the name TFTR and went to Congress for funding, which was granted in January 1975. Conceptual design work was carried out throughout 1975, and detailed design began the next year. TFTR would be the largest tokamak in the world; for comparison, the original ST had a plasma diameter of 12 inches (300 mm), while

352-516: A minimum required value, and the name "Lawson criterion" may refer to this value. On August 8, 2021, researchers at Lawrence Livermore National Laboratory's National Ignition Facility in California confirmed to have produced the first-ever successful ignition of a nuclear fusion reaction surpassing the Lawson's criteria in the experiment. The central concept of the Lawson criterion is an examination of

440-414: A minimum required value, and the name "Lawson criterion" may refer to this value. On August 8, 2021, researchers at Lawrence Livermore National Laboratory's National Ignition Facility in California confirmed to have produced the first-ever successful ignition of a nuclear fusion reaction surpassing the Lawson's criteria in the experiment. The central concept of the Lawson criterion is an examination of

528-547: A new fundamental mode of plasma confinement -- enhanced reversed shear , to reduce plasma turbulence. TFTR remained in use until 1997. It was dismantled in September 2002, after 15 years of operation. It was followed by the NSTX spherical tokamak. Fusion triple product The Lawson criterion is a figure of merit used in nuclear fusion research. It compares the rate of energy being generated by fusion reactions within

616-534: Is 10 times greater than the center of the Sun, and more than enough for breakeven. Unfortunately, to reach these temperatures the triple product had been greatly reduced to 10 , two or three times too small for break-even. Major efforts to reach these conditions simultaneously continued. Donald Grove, TFTR project manager, said they expected to achieve that goal in 1987. This would be followed with D-T tests that would actually produce breakeven, beginning in 1989. Unfortunately,

704-607: Is easy to show that the fusion power is maximized by a fuel mix given by n 1 / n 2 = ( 1 + Z 2 T e / T i ) / ( 1 + Z 1 T e / T i ) {\displaystyle n_{1}/n_{2}=(1+Z_{2}T_{\mathrm {e} }/T_{\mathrm {i} })/(1+Z_{1}T_{\mathrm {e} }/T_{\mathrm {i} })} . The values for n τ {\displaystyle n\tau } , n T τ {\displaystyle nT\tau } , and

792-607: Is easy to show that the fusion power is maximized by a fuel mix given by n 1 / n 2 = ( 1 + Z 2 T e / T i ) / ( 1 + Z 1 T e / T i ) {\displaystyle n_{1}/n_{2}=(1+Z_{2}T_{\mathrm {e} }/T_{\mathrm {i} })/(1+Z_{1}T_{\mathrm {e} }/T_{\mathrm {i} })} . The values for n τ {\displaystyle n\tau } , n T τ {\displaystyle nT\tau } , and

880-431: Is readily obtained: The quantity T 2 ⟨ σ v ⟩ {\displaystyle {\frac {T^{2}}{\langle \sigma v\rangle }}} is also a function of temperature with an absolute minimum at a slightly lower temperature than T ⟨ σ v ⟩ {\displaystyle {\frac {T}{\langle \sigma v\rangle }}} . For

968-431: Is readily obtained: The quantity T 2 ⟨ σ v ⟩ {\displaystyle {\frac {T^{2}}{\langle \sigma v\rangle }}} is also a function of temperature with an absolute minimum at a slightly lower temperature than T ⟨ σ v ⟩ {\displaystyle {\frac {T}{\langle \sigma v\rangle }}} . For

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1056-456: Is the Lawson criterion. For the deuterium – tritium reaction, the physical value is at least The minimum of the product occurs near T = 26 k e V {\displaystyle T=26\,\mathrm {keV} } . A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, nTτ E . For most confinement concepts, whether inertial , mirror , or toroidal confinement,

1144-456: Is the Lawson criterion. For the deuterium – tritium reaction, the physical value is at least The minimum of the product occurs near T = 26 k e V {\displaystyle T=26\,\mathrm {keV} } . A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, nTτ E . For most confinement concepts, whether inertial , mirror , or toroidal confinement,

1232-428: Is the particle density. The volume rate f {\displaystyle f} (reactions per volume per time) of fusion reactions is where σ {\displaystyle \sigma } is the fusion cross section , v {\displaystyle v} is the relative velocity , and ⟨ ⟩ {\displaystyle \langle \rangle } denotes an average over

1320-428: Is the particle density. The volume rate f {\displaystyle f} (reactions per volume per time) of fusion reactions is where σ {\displaystyle \sigma } is the fusion cross section , v {\displaystyle v} is the relative velocity , and ⟨ ⟩ {\displaystyle \langle \rangle } denotes an average over

1408-479: The Department of Energy (DOE) held a large meeting that was attended by all the major fusion labs. Notable among the attendees was Marshall Rosenbluth , a theorist who had a habit of studying machines and finding a variety of new instabilities that would ruin confinement. To everyone's surprise, at this meeting he failed to raise any new concerns. It appeared that the path to break-even was clear. The last step before

1496-701: The Maxwellian velocity distribution at the temperature T {\displaystyle T} . The volume rate of heating by fusion is f {\displaystyle f} times E c h {\displaystyle E_{\mathrm {ch} }} , the energy of the charged fusion products (the neutrons cannot help to heat the plasma). In the case of the D-T reaction, E c h = 3.5 M e V {\displaystyle E_{\mathrm {ch} }=3.5\,\mathrm {MeV} } . The Lawson criterion requires that fusion heating exceeds

1584-603: The Maxwellian velocity distribution at the temperature T {\displaystyle T} . The volume rate of heating by fusion is f {\displaystyle f} times E c h {\displaystyle E_{\mathrm {ch} }} , the energy of the charged fusion products (the neutrons cannot help to heat the plasma). In the case of the D-T reaction, E c h = 3.5 M e V {\displaystyle E_{\mathrm {ch} }=3.5\,\mathrm {MeV} } . The Lawson criterion requires that fusion heating exceeds

1672-459: The TFTR has achieved the densities and energy lifetimes needed to achieve Lawson at the temperatures it can create, but it cannot create those temperatures at the same time. ITER aims to do both. As for tokamaks , there is a special motivation for using the triple product. Empirically, the energy confinement time τ E is found to be nearly proportional to n / P . In an ignited plasma near

1760-413: The TFTR has achieved the densities and energy lifetimes needed to achieve Lawson at the temperatures it can create, but it cannot create those temperatures at the same time. ITER aims to do both. As for tokamaks , there is a special motivation for using the triple product. Empirically, the energy confinement time τ E is found to be nearly proportional to n / P . In an ignited plasma near

1848-404: The D-T reaction, the minimum occurs at T  = 14 keV. The average <σ v > in this temperature region can be approximated as so the minimum value of the triple product value at T  = 14 keV is about This number has not yet been achieved in any reactor, although the latest generations of machines have come close. JT-60 reported 1.53x10 keV.s.m . For instance,

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1936-404: The D-T reaction, the minimum occurs at T  = 14 keV. The average <σ v > in this temperature region can be approximated as so the minimum value of the triple product value at T  = 14 keV is about This number has not yet been achieved in any reactor, although the latest generations of machines have come close. JT-60 reported 1.53x10 keV.s.m . For instance,

2024-425: The Lawson criterion gives a minimum required value for the product of the plasma (electron) density n e and the " energy confinement time " τ E {\displaystyle \tau _{E}} that leads to net energy output. Later analysis suggested that a more useful figure of merit is the triple product of density, confinement time, and plasma temperature T . The triple product also has

2112-425: The Lawson criterion gives a minimum required value for the product of the plasma (electron) density n e and the " energy confinement time " τ E {\displaystyle \tau _{E}} that leads to net energy output. Later analysis suggested that a more useful figure of merit is the triple product of density, confinement time, and plasma temperature T . The triple product also has

2200-468: The above expression into relationship ( 1 ), we obtain This product must be greater than a value related to the minimum of T /<σv>. The same requirement is traditionally expressed in terms of mass density ρ  = < nm i >: Satisfaction of this criterion at the density of solid D-T (0.2 g/cm ) would require a laser pulse of implausibly large energy. Assuming the energy required scales with

2288-423: The above expression into relationship ( 1 ), we obtain This product must be greater than a value related to the minimum of T /<σv>. The same requirement is traditionally expressed in terms of mass density ρ  = < nm i >: Satisfaction of this criterion at the density of solid D-T (0.2 g/cm ) would require a laser pulse of implausibly large energy. Assuming the energy required scales with

2376-503: The alpha particles produced in the deuterium-tritium reactions, which are important for self-heating of the plasma and an important part of any operational design. In 1995, TFTR attained a world-record temperature of 510 million °C - more than 25 times that at the center of the sun. This later was beaten the following year by the JT-60 Tokamak which achieved an ion temperature of 522 million °C (45 keV). Also In 1995, TFTR scientists explored

2464-491: The attack on break-even would be to make a reactor that ran on a mixture of deuterium and tritium , as opposed to earlier machines which ran on deuterium alone. This was because tritium was both radioactive and easily absorbed in the body, presenting safety issues that made it expensive to use. It was widely believed that the performance of a machine running on deuterium alone would be basically identical to one running on D-T, but this assumption needed to be tested. Looking over

2552-443: The cloud and T is the temperature. For his analysis, Lawson ignores conduction losses. In reality this is nearly impossible; practically all systems lose energy through mass leaving the plasma and carrying away its energy. By equating radiation losses and the volumetric fusion rates, Lawson estimated the minimum temperature for the fusion for the deuterium – tritium (D-T) reaction to be 30 million degrees (2.6 keV), and for

2640-443: The cloud and T is the temperature. For his analysis, Lawson ignores conduction losses. In reality this is nearly impossible; practically all systems lose energy through mass leaving the plasma and carrying away its energy. By equating radiation losses and the volumetric fusion rates, Lawson estimated the minimum temperature for the fusion for the deuterium – tritium (D-T) reaction to be 30 million degrees (2.6 keV), and for

2728-687: The confirmation of the Novosibirsk results, they immediately began converting the Model C to a tokamak layout, known as the Symmetrical Tokamak (ST). This was completed in the short time of only eight months, entering service in May 1970. ST's computerized diagnostics allowed it to quickly match the Soviet results, and from that point, the entire fusion world was increasingly focused on this design over any other. During

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2816-416: The density and temperature can be varied over a fairly wide range, but the maximum attainable pressure p is a constant. When such is the case, the fusion power density is proportional to p <σ v >/ T . The maximum fusion power available from a given machine is therefore reached at the temperature T where <σ v >/ T is a maximum. By continuation of the above derivation, the following inequality

2904-416: The density and temperature can be varied over a fairly wide range, but the maximum attainable pressure p is a constant. When such is the case, the fusion power density is proportional to p <σ v >/ T . The maximum fusion power available from a given machine is therefore reached at the temperature T where <σ v >/ T is a maximum. By continuation of the above derivation, the following inequality

2992-526: The density of the electrons alone, but p {\displaystyle p} here refers to the total pressure. Given two species with ion densities n 1 , 2 {\displaystyle n_{1,2}} , atomic numbers Z 1 , 2 {\displaystyle Z_{1,2}} , ion temperature T i {\displaystyle T_{\mathrm {i} }} , and electron temperature T e {\displaystyle T_{\mathrm {e} }} , it

3080-526: The density of the electrons alone, but p {\displaystyle p} here refers to the total pressure. Given two species with ion densities n 1 , 2 {\displaystyle n_{1,2}} , atomic numbers Z 1 , 2 {\displaystyle Z_{1,2}} , ion temperature T i {\displaystyle T_{\mathrm {i} }} , and electron temperature T e {\displaystyle T_{\mathrm {e} }} , it

3168-542: The designs presented at the meeting, the DOE team chose the Princeton design. Bob Hirsch , who recently took over the DOE steering committee, wanted to build the test machine at Oak Ridge National Laboratory (ORNL), but others in the department convinced him it would make more sense to do so at PPPL. They argued that a Princeton team would be more involved than an ORNL team running someone else's design. If an engineering prototype of

3256-404: The deuterium–deuterium (D-D) reaction to be 150 million degrees (12.9 keV). The confinement time τ E {\displaystyle \tau _{E}} measures the rate at which a system loses energy to its environment. The faster the rate of loss of energy, P l o s s {\displaystyle P_{\mathrm {loss} }} , the shorter

3344-404: The deuterium–deuterium (D-D) reaction to be 150 million degrees (12.9 keV). The confinement time τ E {\displaystyle \tau _{E}} measures the rate at which a system loses energy to its environment. The faster the rate of loss of energy, P l o s s {\displaystyle P_{\mathrm {loss} }} , the shorter

3432-412: The early 1970s, Shoichi Yoshikawa was looking over the tokamak concept. He noted that as the size of the reactor's minor axis (the diameter of the tube) increased compared to its major axis (the diameter of the entire system) the system became more efficient. An added benefit was that as the minor axis increased, confinement time improved for the simple reason that it took longer for the fuel ions to reach

3520-438: The energy balance for any fusion power plant using a hot plasma. This is shown below: Net power = Efficiency × (Fusion − Radiation loss − Conduction loss) Lawson calculated the fusion rate by assuming that the fusion reactor contains a hot plasma cloud which has a Gaussian curve of individual particle energies, a Maxwell–Boltzmann distribution characterized by the plasma's temperature. Based on that assumption, he estimated

3608-438: The energy balance for any fusion power plant using a hot plasma. This is shown below: Net power = Efficiency × (Fusion − Radiation loss − Conduction loss) Lawson calculated the fusion rate by assuming that the fusion reactor contains a hot plasma cloud which has a Gaussian curve of individual particle energies, a Maxwell–Boltzmann distribution characterized by the plasma's temperature. Based on that assumption, he estimated

Tokamak Fusion Test Reactor - Misplaced Pages Continue

3696-458: The energy confinement time. It is the energy density W {\displaystyle W} (energy content per unit volume) divided by the power loss density P l o s s {\displaystyle P_{\mathrm {loss} }} (rate of energy loss per unit volume): For a fusion reactor to operate in steady state, the fusion plasma must be maintained at a constant temperature. Thermal energy must therefore be added at

3784-458: The energy confinement time. It is the energy density W {\displaystyle W} (energy content per unit volume) divided by the power loss density P l o s s {\displaystyle P_{\mathrm {loss} }} (rate of energy loss per unit volume): For a fusion reactor to operate in steady state, the fusion plasma must be maintained at a constant temperature. Thermal energy must therefore be added at

3872-445: The energy in the heating systems, this represented a Q of about 0.2, or about only 20% of the requirement for break-even. Further testing revealed significant problems, however. To reach break-even, the system would have to meet several goals at the same time, a combination of temperature, density and the length of time the fuel is confined. In April 1986, TFTR experiments demonstrated the last two of these requirements when it produced

3960-522: The field. Although it became clear that TFTR would not reach break-even, experiments using tritium began in earnest in December 1993, the first such device to move primarily to this fuel. In 1994 it produced a then world-record of 10.7 megawatts of fusion power from a 50-50 D-T plasma (exceeded at JET in the UK, which generated 16MW from 24MW of injected thermal power input in 1997). The two experiments had emphasized

4048-414: The first term, the fusion energy being produced, using the volumetric fusion equation. Fusion = Number density of fuel A × Number density of fuel B × Cross section(Temperature) × Energy per reaction This equation is typically averaged over a population of ions which has a normal distribution . The result is the amount of energy being created by the plasma at any instant in time. Lawson then estimated

4136-414: The first term, the fusion energy being produced, using the volumetric fusion equation. Fusion = Number density of fuel A × Number density of fuel B × Cross section(Temperature) × Energy per reaction This equation is typically averaged over a population of ions which has a normal distribution . The result is the amount of energy being created by the plasma at any instant in time. Lawson then estimated

4224-471: The follow-on PLT design was 36 inches (910 mm), and the TFTR was designed to be 86 inches (2,200 mm). This made it roughly double the size of other large-scale machines of the era; the 1978 Joint European Torus and roughly concurrent JT-60 were both about half the diameter. As PLT continued to generate better and better results, in 1978 and 79, additional funding was added and the design amended to reach

4312-414: The formulas. On the other hand, for cold electrons, the formulas must all be divided by 4 {\displaystyle 4} (with no additional factor for Z > 1 {\displaystyle Z>1} ). Lawson criterion The Lawson criterion is a figure of merit used in nuclear fusion research. It compares the rate of energy being generated by fusion reactions within

4400-433: The fusion fuel to the rate of energy losses to the environment. When the rate of production is higher than the rate of loss, the system will produce net energy. If enough of that energy is captured by the fuel, the system will become self-sustaining and is said to be ignited . The concept was first developed by John D. Lawson in a classified 1955 paper that was declassified and published in 1957. As originally formulated,

4488-433: The fusion fuel to the rate of energy losses to the environment. When the rate of production is higher than the rate of loss, the system will produce net energy. If enough of that energy is captured by the fuel, the system will become self-sustaining and is said to be ignited . The concept was first developed by John D. Lawson in a classified 1955 paper that was declassified and published in 1957. As originally formulated,

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4576-401: The high temperatures of nuclear fusion reactions. The heating current is induced by the changing magnetic fields in central induction coils and exceeds a million amperes. Magnetic fusion devices keep the hot plasma out of contact with the walls of its container by keeping it moving in circular or helical paths by means of the magnetic force on charged particles and by a centripetal force acting on

4664-482: The idea of neutral beam injection . This used small particle accelerators to inject fuel atoms directly into the plasma, both heating it and providing fresh fuel. After a number of modifications to the beam injection system, the newly equipped PLT began setting records and eventually made several test runs at 60 million K, more than enough for a fusion reactor. To reach the Lawson criterion for ignition, all that

4752-483: The inertial case it is more usefully expressed in a different form. A good approximation for the inertial confinement time τ E {\displaystyle \tau _{E}} is the time that it takes an ion to travel over a distance R at its thermal speed where m i denotes mean ionic mass. The inertial confinement time τ E {\displaystyle \tau _{E}} can thus be approximated as By substitution of

4840-483: The inertial case it is more usefully expressed in a different form. A good approximation for the inertial confinement time τ E {\displaystyle \tau _{E}} is the time that it takes an ion to travel over a distance R at its thermal speed where m i denotes mean ionic mass. The inertial confinement time τ E {\displaystyle \tau _{E}} can thus be approximated as By substitution of

4928-508: The knowledge from TFTR to begin studying another approach, the spherical tokamak , in their National Spherical Torus Experiment . The Japanese JT-60 is very similar to the TFTR, both tracing their design to key innovations introduced by Shoichi Yoshikawa (1934-2010) during his time at PPPL in the 1970s. In nuclear fusion, there are two types of reactors stable enough to conduct fusion: magnetic confinement reactors and inertial confinement reactors. The former method of fusion seeks to lengthen

5016-529: The long-sought goal of "scientific breakeven" when the amount of power produced by the fusion reactions in the plasma was equal to the amount of power being fed into it to heat it to operating temperatures. Also known as Q = 1, this is an important step on the road to useful power-producing designs. To meet this requirement, the heating system was upgraded to 50 MW, and finally to 80 MW. Construction began in 1980 and TFTR began initial operations in 1982. A lengthy period of break-in and testing followed. By

5104-467: The losses: Substituting in known quantities yields: Rearranging the equation produces: The quantity T / ⟨ σ v ⟩ {\displaystyle T/\langle \sigma v\rangle } is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product n τ E {\displaystyle n\tau _{E}} . This

5192-467: The losses: Substituting in known quantities yields: Rearranging the equation produces: The quantity T / ⟨ σ v ⟩ {\displaystyle T/\langle \sigma v\rangle } is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product n τ E {\displaystyle n\tau _{E}} . This

5280-468: The mass of the fusion plasma ( E laser ~ ρR ~ ρ ), compressing the fuel to 10 or 10 times solid density would reduce the energy required by a factor of 10 or 10 , bringing it into a realistic range. With a compression by 10 , the compressed density will be 200 g/cm , and the compressed radius can be as small as 0.05 mm. The radius of the fuel before compression would be 0.5 mm. The initial pellet will be perhaps twice as large since most of

5368-468: The mass of the fusion plasma ( E laser ~ ρR ~ ρ ), compressing the fuel to 10 or 10 times solid density would reduce the energy required by a factor of 10 or 10 , bringing it into a realistic range. With a compression by 10 , the compressed density will be 200 g/cm , and the compressed radius can be as small as 0.05 mm. The radius of the fuel before compression would be 0.5 mm. The initial pellet will be perhaps twice as large since most of

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5456-406: The mass will be ablated during the compression. The fusion power times density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burn-up of the fuel is probably more useful. The burn-up should be proportional to the specific reaction rate ( n < σv >) times the confinement time (which scales as T ) divided by

5544-406: The mass will be ablated during the compression. The fusion power times density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burn-up of the fuel is probably more useful. The burn-up should be proportional to the specific reaction rate ( n < σv >) times the confinement time (which scales as T ) divided by

5632-407: The mid-1980s, tests with deuterium began in earnest in order to understand its performance. In 1986 it produced the first 'supershots' which produced many fusion neutrons. These demonstrated that the system could reach the goals of the initial 1976 design; the performance when running on deuterium was such that if tritium was introduced it was expected to produce about 3.5 MW of fusion power. Given

5720-552: The moving particles. By the early 1960s, the fusion power field had grown large enough that the researchers began organizing semi-annual meetings that rotated around the various research establishments. In 1968, the now-annual meeting was held in Novosibirsk , where the Soviet delegation surprised everyone by claiming their tokamak designs had reached performance levels at least an order of magnitude better than any other device. The claims were initially met with skepticism, but when

5808-430: The optimum temperature, the heating power P equals fusion power and therefore is proportional to n T . The triple product scales as The triple product is only weakly dependent on temperature as T . This makes the triple product an adequate measure of the efficiency of the confinement scheme. The Lawson criterion applies to inertial confinement fusion (ICF) as well as to magnetic confinement fusion (MCF) but in

5896-430: The optimum temperature, the heating power P equals fusion power and therefore is proportional to n T . The triple product scales as The triple product is only weakly dependent on temperature as T . This makes the triple product an adequate measure of the efficiency of the confinement scheme. The Lawson criterion applies to inertial confinement fusion (ICF) as well as to magnetic confinement fusion (MCF) but in

5984-568: The outside of the reactor. This led to widespread acceptance that designs with lower aspect ratios were a key advance over earlier models. This led to the Princeton Large Torus (PLT), which was completed in 1975. This system was successful to the point where it quickly reached the limits of its Ohmic heating system, the system that passed current through the plasma to heat it. Among the many ideas proposed for further heating, in cooperation with Oak Ridge National Laboratory , PPPL developed

6072-413: The particle density n : Thus the optimum temperature for inertial confinement fusion maximises <σv>/ T , which is slightly higher than the optimum temperature for magnetic confinement. Lawson's analysis is based on the rate of fusion and loss of energy in a thermalized plasma. There is a class of fusion machines that do not use thermalized plasmas but instead directly accelerate individual ions to

6160-413: The particle density n : Thus the optimum temperature for inertial confinement fusion maximises <σv>/ T , which is slightly higher than the optimum temperature for magnetic confinement. Lawson's analysis is based on the rate of fusion and loss of energy in a thermalized plasma. There is a class of fusion machines that do not use thermalized plasmas but instead directly accelerate individual ions to

6248-581: The power density must be multiplied by the factor ( 1 + Z 1 T e / T i ) ⋅ ( 1 + Z 2 T e / T i ) / 4 {\displaystyle (1+Z_{1}T_{\mathrm {e} }/T_{\mathrm {i} })\cdot (1+Z_{2}T_{\mathrm {e} }/T_{\mathrm {i} })/4} . For example, with protons and boron ( Z = 5 {\displaystyle Z=5} ) as fuel, another factor of 3 {\displaystyle 3} must be included in

6336-581: The power density must be multiplied by the factor ( 1 + Z 1 T e / T i ) ⋅ ( 1 + Z 2 T e / T i ) / 4 {\displaystyle (1+Z_{1}T_{\mathrm {e} }/T_{\mathrm {i} })\cdot (1+Z_{2}T_{\mathrm {e} }/T_{\mathrm {i} })/4} . For example, with protons and boron ( Z = 5 {\displaystyle Z=5} ) as fuel, another factor of 3 {\displaystyle 3} must be included in

6424-479: The pressure forces of the plasma, it seems appropriate to define the effective (electron) density n {\displaystyle n} through the (total) pressure p {\displaystyle p} as n = p / 2 T i {\displaystyle n=p/2T_{\mathrm {i} }} . The factor of 2 {\displaystyle 2} is included because n {\displaystyle n} usually refers to

6512-479: The pressure forces of the plasma, it seems appropriate to define the effective (electron) density n {\displaystyle n} through the (total) pressure p {\displaystyle p} as n = p / 2 T i {\displaystyle n=p/2T_{\mathrm {i} }} . The factor of 2 {\displaystyle 2} is included because n {\displaystyle n} usually refers to

6600-413: The radiation losses using the following equation: P B = 1.4 ⋅ 10 − 34 ⋅ N 2 ⋅ T 1 / 2 W c m 3 {\displaystyle P_{B}=1.4\cdot 10^{-34}\cdot N^{2}\cdot T^{1/2}{\frac {\mathrm {W} }{\mathrm {cm} ^{3}}}} where N is the number density of

6688-413: The radiation losses using the following equation: P B = 1.4 ⋅ 10 − 34 ⋅ N 2 ⋅ T 1 / 2 W c m 3 {\displaystyle P_{B}=1.4\cdot 10^{-34}\cdot N^{2}\cdot T^{1/2}{\frac {\mathrm {W} }{\mathrm {cm} ^{3}}}} where N is the number density of

6776-503: The required energies. The best-known examples are the migma , fusor and polywell . When applied to the fusor, Lawson's analysis is used as an argument that conduction and radiation losses are the key impediments to reaching net power. Fusors use a voltage drop to accelerate and collide ions, resulting in fusion. The voltage drop is generated by wire cages, and these cages conduct away particles. Polywells are improvements on this design, designed to reduce conduction losses by removing

6864-503: The required energies. The best-known examples are the migma , fusor and polywell . When applied to the fusor, Lawson's analysis is used as an argument that conduction and radiation losses are the key impediments to reaching net power. Fusors use a voltage drop to accelerate and collide ions, resulting in fusion. The voltage drop is generated by wire cages, and these cages conduct away particles. Polywells are improvements on this design, designed to reduce conduction losses by removing

6952-534: The results were confirmed by a UK team the next year, this huge advance led to a "virtual stampede" of tokamak construction. In the US, one of the major approaches being studied up to this point was the stellarator , whose development was limited almost entirely to the PPPL. Their latest design, the Model C, had recently gone into operation and demonstrated performance well below theoretical calculations, far from useful figures. With

7040-433: The same rate the plasma loses energy in order to maintain the fusion conditions. This energy can be supplied by the fusion reactions themselves, depending on the reaction type, or by supplying additional heating through a variety of methods. For illustration, the Lawson criterion for the D-T reaction will be derived here, but the same principle can be applied to other fusion fuels. It will also be assumed that all species have

7128-433: The same rate the plasma loses energy in order to maintain the fusion conditions. This energy can be supplied by the fusion reactions themselves, depending on the reaction type, or by supplying additional heating through a variety of methods. For illustration, the Lawson criterion for the D-T reaction will be derived here, but the same principle can be applied to other fusion fuels. It will also be assumed that all species have

7216-451: The same temperature, that there are no ions present other than fuel ions (no impurities and no helium ash), and that D and T are present in the optimal 50-50 mixture. Ion density then equals electron density and the energy density of both electrons and ions together is given, according to the ideal gas law , by where T {\displaystyle T} is the temperature in electronvolt (eV) and n {\displaystyle n}

7304-451: The same temperature, that there are no ions present other than fuel ions (no impurities and no helium ash), and that D and T are present in the optimal 50-50 mixture. Ion density then equals electron density and the energy density of both electrons and ions together is given, according to the ideal gas law , by where T {\displaystyle T} is the temperature in electronvolt (eV) and n {\displaystyle n}

7392-453: The system was unable to meet any of these goals. The reasons for these problems were intensively studied over the following years, leading to a new understanding of the instabilities of high-performance plasmas that had not been seen in smaller machines. A major outcome of TFTR's troubles was the development of highly non-uniform plasma cross-sections, notably the D-shaped plasmas that now dominate

7480-408: The time that ions spend close together in order to fuse them together, while the latter aims to fuse the ions so fast that they do not have time to move apart. Inertial confinement reactors, unlike magnetic confinement reactors, use laser fusion and ion-beam fusion in order to conduct fusion. However, with magnetic confinement reactors you avoid the problem of having to find a material that can withstand

7568-450: The wire cages which cause them. Regardless, it is argued that radiation is still a major impediment. ^a It is straightforward to relax these assumptions. The most difficult question is how to define n {\displaystyle n} when the ion and electrons differ in density and temperature. Considering that this is a calculation of energy production and loss by ions, and that any plasma confinement concept must contain

7656-450: The wire cages which cause them. Regardless, it is argued that radiation is still a major impediment. ^a It is straightforward to relax these assumptions. The most difficult question is how to define n {\displaystyle n} when the ion and electrons differ in density and temperature. Considering that this is a calculation of energy production and loss by ions, and that any plasma confinement concept must contain

7744-434: Was needed was higher plasma density, and there seemed to be no reason this would not be possible in a larger machine. There was widespread belief that break-even would be reached during the 1970s. After the success of PLT and other follow-on designs, the basic concept was considered well understood. PPPL began the design of a much larger successor to PLT that would demonstrate plasma burning in pulsed operation. In July 1974,

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