1 Fusion enhancement due to energy spread of colliding nuclei* 1. Motivation: anomalous electron screening or what else ? 2. Energy spread  fusion enhancement 3. Calculate fusion enhancement due to thermal motion of target atoms 4. Generalize to similar processes: - Lattice vibration - Beam energy width - Energy straggling 5. So far, so what? * Work in progress by B Ricci, F Villante and G Fiorentini

2 Anomalous electron screening or what else? At the lowest measured energies, fusion rates are found generally larger than expected, i.e. the enhancement factor f=  meas /  Bare is: f > f ad where f ad is the maximal screening effect consistent with QM [ In the adiabatic limit energy transfer from electrons to colliding nuclei is maximal ]. Measured values are typically: (f-1)  ( ) % for p+d, d+d, d+ 3 He in gas target (f-1)  100% for d+d with d inplanted in metals Are we seeing anomalous screening or something else?

3 Energy spread implies fusion enhancement Consider an example: -The projectile has fixed V -The target has a velocity distribution due (e.g.) to thermal motion: P(v)  exp[-v 2 / 2v t 2 ] -Can one neglect the target velocity distribution assuming that “it is zero on average”, i.e.: =  (V) ??? NO:Due to strong velocity dependence, approaching particles have larger weight than the receding ones.  i f= /  (V) >1 V V+v t V-v t

4 Calculation of the effect Generally one has to calculate: f= exp [ v o v t 2 / 2V 4 ] f= /  (V) >1 Where is the average over the velocity distribution: f =  dv exp [-v o /  V-v  -v 2 / 2v t 2 ] / exp[- v o /V]  dv exp [-v 2 / 2v t 2 ] The calculation can be easily done by a “Gamow trick”. The result is*: where V=proj. vel., v t = target av. thermal vel.,v o =2  Z 1 Z 2  c * We are assuming S=const; for S=S(E) see later…The result is to the leading order in v/V  Maxwell vv max

5 Remarks 1. f> 1 i.e. always enhancement Gamow-like peak; largest contribution from target nuclei with: v = v max = v o v t 2 / V 2 f is strongly energy dependent, f  exp (-k/ E 2 ). En. Dependence different fromscreening: f sc  exp (-k/ E 3/2 ) Effect is small in present conditions: (f-1)  for d+d at (c.m.) E = 2 keV It would be significant at extremely low energies: (f-1)  10% for d+d at E = 0.2 keV f= exp [ v o 2 v t 2 / 2V 4 ] V =proj. vel., v t = target av. thermal vel. v o =2  Z 1 Z 2  c

6 Effective energy enhancement If one takes into account that S=S(E) an additional effect arises. The “Gamow peak”means that the S factor is measured for an effective velocity V eff = V(1+ v o v t 2 / V 3) Equivalently, the effective c.m. energy E eff is larger than E=1/2  V 2 : E eff =E (1 + v o v t 2 (  /2E) 3/2 ] 2. The effect can thus be interpreted as an enhancement of the effective collision energy. Really a very tiny effect: for d+d at E CM = 2 keV  Maxwell vv max S

7 Correction of S If S exp has been measured at a nominal energy E=1/2  V 2, from: S exp (E) =  exp  E(V  exp (v o /V) in order to obtain the true S factor one has to: - Change to the effective energy E--> E eff -Apply a renormalization factor: S(E eff ) = S exp (E) E exp /E exp [-v o 2 v t 2 / 2V(E) 4 ] S exp (E) S(E eff ) E E eff

8 Generalization to similar processes There are two ingredients in the calculation:   exp [ - v o /V rel  V rel =relative velocity) P(v)  exp [-v 2 / 2v t 2 ] The same scheme can be used for other processes, which produce a (Gaussian) velocity spread of target and/or projectile nuclei. One only has to re-interpret v t 2, by introducing a suitable (v t 2 ) eff f= exp [ v o 2 (v t 2 ) eff / 2V 4 ]

9 Vibrational effects Consider the target nucleus (e.g. d) inplanted in a crystal*). The typical vibrational energies are E vib =(0.1-1) eV Since the collision time is short compared to the vibrational period, one can use the sudden approximation for the target nucleus motion. In the harmonic oscillator approximation, one has: = 1/2 E vib This means = 1/2 (KT) eff  (V t 2 ) eff = E vib /m d d *)Similar considerations hold for molecular vibrations

10 Enhancement due Vibrational effects Resulting effects are small: (f-1)  3 ( )for d+d at E CM = 2 keV They can become significant at smaller energy. The value observed for d in some metals, f-1  100%, would correspond to E vib  10 eV d This gives for the enhancement factor: f= exp [ v o 2  vib / 2 m d V 4 ]

11 Beam energy width The produced beam is not really monochromatic: P(E)  exp [-(E-E L ) 2 /2   beam ] For LUNA,  beam  10 eV. (V t 2 ) eff =   beam / 2m p E L This can be transformed into an approximately gaussian velocity distribution with:  EELEL ( m p = projectile mass)

12 Enhancement due to Beam energy width Effects are very small in the LUNA condition: (f-1)  for d+d at E CM = 2 keV and  beam  10 eV The effect behaves quadratically with  beam and it can be significant if momentum resolution is worse.  EELEL The enhancement factor is thus: f= exp [ v o 2   beam / 2 m 2 p V 6 ]

13 Check As a check, one can show that the same result can be obtained by integrating  directly over the energy distribution : P(E)  exp [-(E-E L ) 2 /2   beam ] f = /  (E L )= exp( v o 2   beam / 2m p 2 V 6 ) By using the saddle point method and assuming  beam << E L one finds:  EELEL This is the same result as before.

14 Energy loss and straggling Due to atomic collisions in the target one has - Energy loss, E lost - Energy Straggling,  stra  stra =(E lost  ) 1/2 If  is the energy lost in each of N collisions E lost  stra    For E lost  1KeV,   10 eV  stra  100 eV. One has the same formula as before, however with  stra >>  beam : E f = /  (E L )= exp( v o 2 E lost  / 2m p 2 V 6 )

15 Competition between energy loss and straggling Consider particles entering the target with kin. energy E in. As they advance their kin. energies are decreased. When the average kin. energy is E L =1/2 m p V 2 the correct weight to the cross section is: 1/V 2 exp (v o /V) exp( v o 2 E lost  / 2m p 2 V 6 ) The last term is due to straggling. It is: -negliglible at E L  E in ( since E lost =0), where most fusions occur. -large at small energies, when fusion is anyhow suppressed. For this reason we expect the effect is not important... E

16 So far, so what ? Energy spread is a mechanism which provides fusion enhancement We have found a general expression for calculating the enhancement f= /  due to a gaussian spread: Quantitative estimates for d+d at E cm = 2KeV: thermal(f-1)  vibrational(f-1)  beam (f-1)  No explanation for anomalous screening found; actually we can exclude several potential candidates. f= exp [ v o 2 (v t 2 ) eff / 2V 4 ]