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Eric Prebys, FNAL

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 2 Consider two particles in a bunch. Below transition turn n turn n+1 Further apart Above transition… turn n turn n+1 Closer together That is, the particles behave as if they had “negative mass” Consider a beam of uniform line density

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 3 The fields outside the beam are given by Inside the beam, the enclosed current/charge scales as r 2 /a 2, so We now find the field along the beam axis using

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 4 Assume we have the beam propagating through a beam pipe of radius b Note, λ not constant

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 5 Any perturbation is propagating with the beam, so

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 6 If the wall is perfectly conducting, then E w =0, and we have We’ll factor any perturbations in the line density into harmonic components azimuthal location frequency of oscillation n=mode number. General solution will be a combination of these. Mode will propagate with an angular frequency phase velocity of perturbation around ring

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 7 Recall, λ is a charge density, so it must satisfy the continuity equation

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 8 Look for a solution of the form Assume small

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 9 We can write the current in the form We now consider an individual particle in the distribution particle deformation But the angular velocity of an individual particle around the ring is related to the period by slip factor

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 10 So we can write We’re now going to introduce the concept of “longitudinal impedance to characterize the energy lost per particle in terms of the total current, defined by We’re only interested in the fluctuating part, so we write recall Combining, we get

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 11 Substitute and we get Recall so

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 12 So we have Motion will be stable if RHS is both real and positive, so energy loss given by fraction of total set but

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USPAS, Knoxville, TN, Jan. 20-31, 2014 Lecture 16 -Negative Mass Instability 13 so we have imaginary and negative so for motion to be stable, we want η<0 In other words, motion will only be stable below transition. This is why unbunched beams are not stable above transition.

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