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Low temperature scattering of helium and hydrogen
PflYSICA
Physica B 194-196 (1994) 887-888 North-Holland
L O W T E M P E R A T U R E SCATTERING OF HELIUM AND HYDROGEN John Mester,* Eric Meyer, Tito Huber, Meritt Reynolds, and Isaac Silvera Dept. of Physics, Harvard University, Cambridge, MA 02138 We investigate two body interactions of 4He and atomic H in the low energy limit. H-4He and 4He-4He integral scattering cross sections are determined from the attenuation of beams of 4He atoms through target gases of H and 4He at temperatures below 400 mK. We analyze the results within the Quantum Corresponding States framework. Implications for scattering isotopic species of He and H are discussed. 1. INTRODUCTION In this paper we present H-4He scattering measurements made in the low temperature regime where scattering processes require a full quantum mechanical treatment. We compare the results with measurements of 4He-4He scattering made under similar conditions.Ill The experiment probes the HHe potential, which calculations predict to be one of the most shallow interatomic potentials[2]; only the H-H triplet and He-He potentials have comparable well depth. New methods have enabled us to measure the H-4He absolute total scattering cross section over a background gas temperature range from 175 to 400 mK. Collision energies are sufficiently low that the measurements probe an extensive range of the interaction potential including the well and the long range tail. The scattering is dominated by the S-wave cross section. 2. EXPERIMENT
The experimental apparatus consists of a cell with two bolometers, -a heater and a detectorseparated by a distance of 8.3 cm. 4He is added to form a saturated film on the cell walls. First, the cell is stabilized at a fixed temperature and heatpulse beams of 4He are created and detected. Then, spin-polarized H is loaded into the cell and the measurements are repeated yielding an attenuated time of flight signal due to scattering of the beam by H atoms. Attenuation measurements have been made with several H densities at cell temperatures of 175, 200, 250, 300, 350, and 400 mK. Low heater powers are used so that the signals are Boltzmann time of flight distributions.[3] Peak 4He beam velocities range from 38 to 58 m/s. The H
12 ¢,1
s 6
O
o 0.15
!
!
I
I
1
0.20
0.25
0.30
0.35
0.40
Temperature(K) Figure 1. H-He effective cross section vs. H temperature. Calculation based on the R2 potential
is given by the solid line. density is determined by triggering the atoms to recombine into molecular 1-12 and measuring the total recombination energy. We determine the scattering cross sections by fitting time of flight signals for each cell temperature and H density. We first simulate the signal with no H present by calculating the detector response using heater and detector characteristics which are measured independently. A good match to the data is obtained by varying one overall scale factor. We then analyze the attenuation data by including an attenuation factor, given by exp(-nHCL), where nH is the H density and L is the heater-detector distance, and then varying the cross section or, until a good fit is obtained. The results are shown in figure 1. The error bars represent uncertainties in corrections made to account for the finite size of the heater and detector and the influence of the H background on detector
*Present Address: Hansen Experimental Physics Lab, Stanford University, Stanford, CA 94305-4085 0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1024-G
888 response as well as the intrinsic error in the H density determination. Additional systematic effects may be present in the 350 and 400 mK data due to the significant 4 H e vapor at these temperatures.[ 1] In order to compare measurements with theory, we have calculated total scattering cross sections for H-4He collisions as a function of relative velocity, and then averaged over the velocity distribution of the 4He beam and of H at the cell temperature. The results are shown for the R2 potential of Jochemsen et al.[2] Using this potential we find that the scattering length equals -0.11 A, which gives a zero energy scattering cross section of cr = 0.15/~ 2. Deviations from calculations can be seen in the 350 and 400 mK data. However, the most striking feature of the results is the overall size of the cross section; the measurements confirm the very small values of the low temperature cross section for H-4He. 3. DISCUSSION Classically, from a molecular diameter approximation, one would expect the scattering cross section to be on the order of m "2, where r is the value of the zero crossing of the potential. For H-He this value is 3.2 A so the classical cross section is about 32 ,/k2, much larger than measured. The classical prediction for the He-He cross section is about 22 A 2. In this case, the classical value is much smaller then our 4He-4He measurements, which are several hundred ]k2 at 300 - 400 mK.[1] Thus, the heat pulse technique has enabled us to measure scattering cross sections at two different quantum mechanical extremes. These low temperature scattering results can be examined within a unified framework in terms of the quantum parameter, rl, used in the quantum theory o f corresponding states.[4,5] If we parameterize the H-He and He-He potentials in terms of the well depth, e, and zero crossing, s, the quantum parameter is defined ~l -- tr2/2~tes2 where ~t is the reduced mass. Thus, r I is a ratio of the zero point energy to the potential strength. Treating r I as a continuous variable we plot the scattering length as a function of rl-1 in figure 2. The change from _oo to +o,, indicates the existence of a two body bound state.[5] For 4He-4He scattering we measure a very large scattering cross section due to a nearly divergent scattering length, while for H-4He
100
- ~ K
"<~ 5 0 -=
3He_3He
~ "5 ~ -50 r.o -1 o o I0
H_4~H~3He-4He
1
I
I
I
2
4
6
8
rl
-I
0
Figure 2. Scattering length vs. the inverse of the quantum parameter. Arrows indicate approximate values for several two body systems. After ref. [5]. scattering we measure a very small scattering cross section due to a nearly zero scattering length. In both cases the extremes can be seen to arise from incidental tunings of the reduced masses and potential strengths. The nearly divergent scattering length occurs because the reduced mass of the 4He4He system produces a bound state near the continuum. The nearly zero scattering length for H-4He results because the reduced mass is near the cross-over value where the interaction changes from being generally repulsive to generally attractive. We see from this qualitative analysis that collisions of different isotopic species, such as 3He-4He and 3He-3He, and of H-3He, D-4He, etc. should display entirely different scattering dynamics than 4He-4He and H-4He, even though the respective potentials are essentially identical. Scauering measurements of some of these isotopic combinations are under way,J6] We acknowledge the support of DOE grant DE-FG021-85ER45190. REFERENCES 1.
2. 3. 4. 5. 6.
J.C. Mester, E.S. Meyer, T.E. Huber, M. W. Reynolds, and I.F. Silvera, J. Low Temp. Phys. 89, 569 (1992). and E.S. Meyer, J.C. Mester, B.Freedman, J. Kim, M.W. Reynolds, Z. Zhao, and I.F. Silvera, submitted to LT20 (1993). R. Jochemsen, A. J. Berlinsky, and W. N. Hardy, Can. J. Phys. 62751 (1984). E.S. Meyer, J.C. Mester, and I.F. Silvera, Phys. Rev. Lett. 70, 908-11 (1993). L.H. Nosanow, J. Phys. 41 C7-1 (1980). Y.H. Uang and W.C. Stwalley, J. Chem. Phys. 76 5069 (1982). To be published by present authors.
Physica B 194-196 (1994) 887-888 North-Holland
L O W T E M P E R A T U R E SCATTERING OF HELIUM AND HYDROGEN John Mester,* Eric Meyer, Tito Huber, Meritt Reynolds, and Isaac Silvera Dept. of Physics, Harvard University, Cambridge, MA 02138 We investigate two body interactions of 4He and atomic H in the low energy limit. H-4He and 4He-4He integral scattering cross sections are determined from the attenuation of beams of 4He atoms through target gases of H and 4He at temperatures below 400 mK. We analyze the results within the Quantum Corresponding States framework. Implications for scattering isotopic species of He and H are discussed. 1. INTRODUCTION In this paper we present H-4He scattering measurements made in the low temperature regime where scattering processes require a full quantum mechanical treatment. We compare the results with measurements of 4He-4He scattering made under similar conditions.Ill The experiment probes the HHe potential, which calculations predict to be one of the most shallow interatomic potentials[2]; only the H-H triplet and He-He potentials have comparable well depth. New methods have enabled us to measure the H-4He absolute total scattering cross section over a background gas temperature range from 175 to 400 mK. Collision energies are sufficiently low that the measurements probe an extensive range of the interaction potential including the well and the long range tail. The scattering is dominated by the S-wave cross section. 2. EXPERIMENT
The experimental apparatus consists of a cell with two bolometers, -a heater and a detectorseparated by a distance of 8.3 cm. 4He is added to form a saturated film on the cell walls. First, the cell is stabilized at a fixed temperature and heatpulse beams of 4He are created and detected. Then, spin-polarized H is loaded into the cell and the measurements are repeated yielding an attenuated time of flight signal due to scattering of the beam by H atoms. Attenuation measurements have been made with several H densities at cell temperatures of 175, 200, 250, 300, 350, and 400 mK. Low heater powers are used so that the signals are Boltzmann time of flight distributions.[3] Peak 4He beam velocities range from 38 to 58 m/s. The H
12 ¢,1
s 6
O
o 0.15
!
!
I
I
1
0.20
0.25
0.30
0.35
0.40
Temperature(K) Figure 1. H-He effective cross section vs. H temperature. Calculation based on the R2 potential
is given by the solid line. density is determined by triggering the atoms to recombine into molecular 1-12 and measuring the total recombination energy. We determine the scattering cross sections by fitting time of flight signals for each cell temperature and H density. We first simulate the signal with no H present by calculating the detector response using heater and detector characteristics which are measured independently. A good match to the data is obtained by varying one overall scale factor. We then analyze the attenuation data by including an attenuation factor, given by exp(-nHCL), where nH is the H density and L is the heater-detector distance, and then varying the cross section or, until a good fit is obtained. The results are shown in figure 1. The error bars represent uncertainties in corrections made to account for the finite size of the heater and detector and the influence of the H background on detector
*Present Address: Hansen Experimental Physics Lab, Stanford University, Stanford, CA 94305-4085 0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1024-G
888 response as well as the intrinsic error in the H density determination. Additional systematic effects may be present in the 350 and 400 mK data due to the significant 4 H e vapor at these temperatures.[ 1] In order to compare measurements with theory, we have calculated total scattering cross sections for H-4He collisions as a function of relative velocity, and then averaged over the velocity distribution of the 4He beam and of H at the cell temperature. The results are shown for the R2 potential of Jochemsen et al.[2] Using this potential we find that the scattering length equals -0.11 A, which gives a zero energy scattering cross section of cr = 0.15/~ 2. Deviations from calculations can be seen in the 350 and 400 mK data. However, the most striking feature of the results is the overall size of the cross section; the measurements confirm the very small values of the low temperature cross section for H-4He. 3. DISCUSSION Classically, from a molecular diameter approximation, one would expect the scattering cross section to be on the order of m "2, where r is the value of the zero crossing of the potential. For H-He this value is 3.2 A so the classical cross section is about 32 ,/k2, much larger than measured. The classical prediction for the He-He cross section is about 22 A 2. In this case, the classical value is much smaller then our 4He-4He measurements, which are several hundred ]k2 at 300 - 400 mK.[1] Thus, the heat pulse technique has enabled us to measure scattering cross sections at two different quantum mechanical extremes. These low temperature scattering results can be examined within a unified framework in terms of the quantum parameter, rl, used in the quantum theory o f corresponding states.[4,5] If we parameterize the H-He and He-He potentials in terms of the well depth, e, and zero crossing, s, the quantum parameter is defined ~l -- tr2/2~tes2 where ~t is the reduced mass. Thus, r I is a ratio of the zero point energy to the potential strength. Treating r I as a continuous variable we plot the scattering length as a function of rl-1 in figure 2. The change from _oo to +o,, indicates the existence of a two body bound state.[5] For 4He-4He scattering we measure a very large scattering cross section due to a nearly divergent scattering length, while for H-4He
100
- ~ K
"<~ 5 0 -=
3He_3He
~ "5 ~ -50 r.o -1 o o I0
H_4~H~3He-4He
1
I
I
I
2
4
6
8
rl
-I
0
Figure 2. Scattering length vs. the inverse of the quantum parameter. Arrows indicate approximate values for several two body systems. After ref. [5]. scattering we measure a very small scattering cross section due to a nearly zero scattering length. In both cases the extremes can be seen to arise from incidental tunings of the reduced masses and potential strengths. The nearly divergent scattering length occurs because the reduced mass of the 4He4He system produces a bound state near the continuum. The nearly zero scattering length for H-4He results because the reduced mass is near the cross-over value where the interaction changes from being generally repulsive to generally attractive. We see from this qualitative analysis that collisions of different isotopic species, such as 3He-4He and 3He-3He, and of H-3He, D-4He, etc. should display entirely different scattering dynamics than 4He-4He and H-4He, even though the respective potentials are essentially identical. Scauering measurements of some of these isotopic combinations are under way,J6] We acknowledge the support of DOE grant DE-FG021-85ER45190. REFERENCES 1.
2. 3. 4. 5. 6.
J.C. Mester, E.S. Meyer, T.E. Huber, M. W. Reynolds, and I.F. Silvera, J. Low Temp. Phys. 89, 569 (1992). and E.S. Meyer, J.C. Mester, B.Freedman, J. Kim, M.W. Reynolds, Z. Zhao, and I.F. Silvera, submitted to LT20 (1993). R. Jochemsen, A. J. Berlinsky, and W. N. Hardy, Can. J. Phys. 62751 (1984). E.S. Meyer, J.C. Mester, and I.F. Silvera, Phys. Rev. Lett. 70, 908-11 (1993). L.H. Nosanow, J. Phys. 41 C7-1 (1980). Y.H. Uang and W.C. Stwalley, J. Chem. Phys. 76 5069 (1982). To be published by present authors.